swanhild bernstein uwe kähler irene sabadini frank sommen

Trends in Mathematics

Swanhild BernsteinUwe KählerIrene SabadiniFrank SommenEditors

Hypercomplex Analysis: New Perspectives and Applications

Trends in Mathematics Trends in Mathematics is a series devoted to the publication of volumes arising from conferences and lecture series focusing on a particular topic from any area of mathematics. Its aim is to make current developments available to the community as rapidly as possible without compromise to quality and to archive these for reference. Proposals for volumes can be submitted using the Online Book Project SubmissionForm at our website www.birkhauser-science.com. Material submitted for publication must be screened and prepared as follows: All contributions should undergo a reviewing process similar to that carried out by journals and be checked for correct use of language which, as a rule, is English. Articles without proofs, or which do not contain any significantly new results, should be rejected. High quality survey papers, however, are welcome. We expect the organizers to deliver manuscripts in a form that is essentially ready for direct reproduction. Any version of TEX is acceptable, but the entire collection of files must be in one particular dialect of TEX and unified according to simple instructions available from Birkhäuser. Furthermore, in order to guarantee the timely appearance of the proceedings it is essential that the final version of the entire material be submitted no later than one year after the conference.

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Swanhild Bernstein Uwe Kähler

Editors

Hypercomplex Analysis: NewPerspectives and Applications

Irene SabadiniFrank Sommen

Editors Swanhild Bernstein Uwe Kähler Institute of Applied Analysis Departamento de Matemática TU Bergakademie Freiberg Universidade de Aveiro Freiberg, Germany Aveiro, Portugal Irene Sabadini Frank Sommen Dipartimento di Matematica Dept. Mathematical Analysis Politecnico di Milano University of Gent Milano, Italy Gent, Belgium

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ISSN 2297-0215 ISSN 2297-024X (electronic) ISBN 978-3-319-08770-2 ISBN 978-3-319-08771-9 (eBook) DOI 10.1007/978-3-319-08771-9 Springer Cham Heidelberg New York Dordrecht London

Mathematics Subject Classification (2010): 30G35, 30G25, 22E46, 32A50, 68U10

Library of Congress Control Number: 2014952606

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Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

R. Abreu Blaya, J. Bory Reyes, A. Guzman Adan and U. KahlerSymmetries and Associated Pairs in Quaternionic Analysis . . . . . . . . . . 1

D. Alpay, F. Colombo and I. SabadiniGeneralized Quaternionic Schur Functions in the Ball andHalf-space and Krein–Langer Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . 19

D. Alpay, F. Colombo, I. Sabadini and G. SalomonThe Fock Space in the Slice Hyperholomorphic Setting . . . . . . . . . . . . . . 43

E. Ariza and A. Di TeodoroMulti Mq-monogenic Function in Different Dimension . . . . . . . . . . . . . . . 61

S. BernsteinThe Fractional Monogenic Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

L.J. Carmona L., L.F. Resendis O. and L.M. Tovar S.Weighted Bergman Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

D. Eelbode and N. VerhulstOn Appell Sets and Verma Modules for sl (2) . . . . . . . . . . . . . . . . . . . . . . . 111

S.-L. Eriksson, H. Orelma and N. VieiraIntegral Formulas for k-hypermonogenic Functions in R3 . . . . . . . . . . . . 119

R. Ghiloni, V. Moretti and A. PerottiSpectral Properties of Compact Normal Quaternionic Operators . . . . . 133

Yu. Grigor’evThree-dimensional Quaternionic Analogue of theKolosov–Muskhelishvili Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

K. Gurlebeck and D. LegatiukOn the Continuous Coupling of Finite Elementswith Holomorphic Basis Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

vi Contents

K. Gurlebeck and H. Manh NguyenOn ψ-hyperholomorphic Functions anda Decomposition of Harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

U. Kahler and N. VieiraFractional Clifford Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

Y. KrasnovSpectral Properties of Differential Equations in Clifford Algebras . . . . 203

D.C. Struppa, A. Vajiac and M.B. VajiacDifferential Equations in Multicomplex Spaces . . . . . . . . . . . . . . . . . . . . . . 213

Preface

At the 9th International ISAAC Congress (International Society for Analysis, itsApplications, and Computations), held at the Pedagogical University of Krakow,Krakow, Poland from August 5 to August 9, 2013, one of the largest sessions wason “Clifford and Quaternionic Analysis” with around 40 speakers coming fromall parts of the world: Belgium, Cech Republic, China, Finland, Germany, Israel,Italy, Mexico, Portugal, Russia, Turkey, Venezuela, Ukraine, United Kingdom andthe United States.

While there are official congress proceedings, the success of the session ledthe organizers to ask the participants to present their most recent and promisingachievements in a special volume to promote the exciting field of hypercomplexanalysis. This volume contains a careful selection of 15 of these papers whichcover several different aspects of hypercomplex analysis going from function theoryover quaternions, Clifford numbers and multicomplex numbers, operator theory,monogenic signals, to the recent field of fractional Clifford analysis. Additionally,applications to image processing, crack analysis, and the theory of elasticity arecovered. All contributed papers represent the most recent achievements in thearea. We hope that anybody interested in the field can find many new ideas andpromising new directions in these papers.

The Editors are grateful to the contributors to this volume and to the referees,for their painstaking and careful work. They also would like to thank the Peda-gogical University in Krakow for hosting the Conference and Vladimir Mityushev,in particular, as Chairman of the local organising committee.

May 2014, Swanhild BernsteinUwe Kahler

Irene SabadiniFrank Sommen


Trends in Mathematics, 1–18c© 2014 Springer International Publishing Switzerland

Symmetries and Associated Pairsin Quaternionic Analysis

Ricardo Abreu Blaya, Juan Bory Reyes, Alı Guzman Adanand Uwe Kahler

Abstract. The present paper is aimed at proving necessary and sufficient con-ditions on the quaternionic-valued coefficients of a first-order linear operatorto be associated to the generalized Cauchy–Riemann operator in quarternionicanalysis and explicitly we give the description of all its nontrivial first-ordersymmetries.

Mathematics Subject Classification (2010). 30G35.

Keywords. Quaternionic analysis, generalized Cauchy–Riemann operator,symmetries and associated pairs.

1. Motivation and basic facts of quaternionic analysis

Approaches by symmetry operators and methods based on associated pairs notonly play an important role for finding explicit solutions to systems of partial dif-ferential equations, see for instance [5, 9, 10, 11, 15, 16, 17], but are also closelylinked to invariance groups of operators. One of the important points is that first-order symmetries form a Lie algebra where the action of the transformation groupis induced by the Lie derivatives [12]. This was used quite extensively in the past inClifford Analysis [23, 21, 22, 3]. Furthermore, the study of first-order symmetriesof the Cauchy–Riemann–Fueter operator, as well as the description of all its asso-ciated pairs, has been done recently by Y. Krasnov [9] and by T.V. Nguyen [17].

Quaternionic analysis offers a function theory related to Cauchy–Riemann–Fueter operator, which represent a generalization of classical complex analysis to

This work was supported by Portuguese funds through the CIDMA – Center for Researchand Development in Mathematics and Applications, and the Portuguese Foundation for Sci-

ence and Technology (“FCT – Fundacao para a Ciencia e a Tecnologia”), within project PEst-OE/MAT/UI4106/2014.

2 R. Abreu Blaya, J. Bory Reyes, A. Guzman Adan and U. Kahler

higher-dimensional Euclidean space. For a more detailed account on this matteralong more traditional lines we refer the reader to [2, 7, 8].

Starting from a definition of a generalized Cauchy–Riemann operator (relatedto some fixed orthonormal basis in R4) proposed by Vasilevski and Shapiro ([24])quaternionic analysis shows important advantages in the effort of solving partialdifferential equations in the hypercomplex framework. Their approach is based onthe notion of a structural set ψk, representing a general orthonormal frame. Whilethe obtained function theory is in most aspects the same as for the generalizedCauchy–Riemann operator with respect to the standard basis this notion is thestarting point for addressing problems which arise when more than one structuralset is involved. Such cases arise, for example, when one studies derivatives ofmonogenic functions with respect to a different orthonormal frame. While in thequaternionic setting Mitelman/Shapiro [13] showed that the conjugate generalizedCauchy–Riemann operator can play the role of a ψ-hyperholomorphic derivative, itcannot separate all directions (which later led to the notion of a hyperholomorphicconstant), which means that to recover all partial derivatives like in the complexcase one would need three differential operators where each of them is obtainedby conjugating only one of the elements of the structural set.

Another example is the study of Beltrami equations, in particular the studyof the monogenic part of its solution. Hereby, one major problem arises. Whileone structural set can be transformed into another by an orthogonal transfor-mation, in general this transformation will not be a rotation. But a generalizedCauchy–Riemann operator itself is only invariant under rotations, not under gen-eral orthogonal transformations. This leads to the notion of left- (right-) equiv-alent structural sets as the subclass of structural sets which can be transformedinto each other by an appropriate rotation. This leads to such interesting proper-ties that while for left- (right-) equivalent structural sets the corresponding sets ofψ-hyperholomorphic functions are coinciding that is not anymore true for generalstructural sets.

Additionally, while in the framework of the quaternionic algebra it is naturalto represent rotations as multiplications of quaternions, this is not possible, forexample, in case of a reflection. Therefore, the treatment of general orthogonaltransformations requires the embedding of the quaternions in a higher-dimensionalalgebra. This can be done either by considering quaternions as 4× 4-dimensionalreal matrices and applying a general orthogonal transformation to this matrix orby embedding the quaternions into a higher-dimensional algebra with signature(4, 4). In both cases the resulting algebra of endomorphisms has to be isomorphicto the full matrix algebra of 4×4-matrices, not just the sub-algebra of quaternions.

The present paper is aimed at proving necessary and sufficient conditions onthe quaternionic-valued coefficients of a first-order linear operator to be associatedto the generalized Cauchy–Riemann operator in quarternionic analysis and weexplicitly describe all its nontrivial first-order symmetries. This is done at first bylinking with matrix description of our operator. In the last section we will showthat the traditional approach by Sommen/van Acker [21, 22] via identifying the

Symmetries and Associated Pairs in Quaternionic Analysis 3

algebra of endomorphisms with a complexified Clifford algebra does also work inthis case. This work can be regarded as a continuation of that in [1], where theauthors deal with simultaneous null solutions of two different generalized Cauchy–Riemann operators.

In the following, we review briefly the basic facts of quaternionic analysisneeded throughout the paper.

Let H be the set of real quaternions with a unit (denoted by 1), generatedby {i, j, k}. This means that any element x from H is of the form x = x0 + x1i+x2j + x3k, where xm ∈ R,m ∈ N3 ∪ {0}; N3 := {1, 2, 3}.

In this paper we denote the generators by 1 =: e0, i =: e1, j =: e2, andk =: e3 subject to the multiplication rules

e2m = −1, m ∈ N3,

e1 e2 = −e2 e1 = e3; e2 e3 = −e3 e2 = e1; e3 e1 = −e3 e1 = e2,

more suited for a future extension.

If x =

3∑m=0

xm em is a quaternion then �x :=

⎛⎜⎜⎝x0

x1

x2

x3

⎞⎟⎟⎠ = (x0, x1, x2, x3)T ∈ R4,

where the index T denotes transposition.

With natural operations of addition and multiplication H is a non-commuta-tive, associative skew-field. There is the quaternionic conjugation, which plays animportant role and is defined as follows:

em := −em, ∀m ∈ N3.

This involution extends onto the whole H as an R-linear mapping: If x ∈ H then

x := x0 − x1e1 − x2e2 − x3e3.

We have

x · y := y · x and x · x = x · x = |x|2 ∈ R.

This norm of a quaternion coincides with the usual Euclidean norm in R4. There-fore, for x ∈ H \ {0}, the quaternion x−1 := x/|x|2 is a multiplicative inverse of x.

The quaternion x · y, coincides with the result of multiplying �y by the leftregular matrix representation of x given by:⎛⎜⎜⎝

x0 −x1 −x2 −x3

x1 x0 −x3 x2

x2 x3 x0 −x1

x3 −x2 x1 x0

⎞⎟⎟⎠ ·⎛⎜⎜⎝

y0y1y2y3

⎞⎟⎟⎠ =: Bl(x) · �y.

We will denote by Bl := {Bl(x) : x ∈ H} the set of all matrices of left regularrepresentations of real quaternions.


One can compute directly that:

• Bl(x + y) = Bl(x) +Bl(y),• Bl(xy) = Bl(x)Bl(y),• Bl(e0) = I4, the identity matrix 4× 4,• Bl(λx) = λBl(x), λ ∈ R,• Bl(x) = Bl(x)

T ,• detBl(x) = |x|4.

In this way,

x ∈ H �→ Bl(x) ∈ Bl (1.1)

is an isomorphism of real algebras.

We consider functions f defined in a domain Ω ⊂ R4 and taking values inH. Such a function may be written as f = f0 + f1e1 + f2e2 + f3e3 and each timewe assign a property such as continuity, differentiability, integrability, and so on,to f it is meant that all R-components fm share this property. Thus notationsf ∈ Cp(Ω,H), p ∈ N ∪ {0}, will have the usual component-wise meaning.

Let Mn×m(R) be the set of real n ×m matrix, (n,m ∈ N). We can identifyany f : Ω→ H with two specific matrix functions:

�f :=

⎛⎜⎜⎝f0f1f2f3

⎞⎟⎟⎠ : Ω→M4×1(R)

and

Bl(f) :=

⎛⎜⎜⎝f0 −f1 −f2 −f3f1 f0 −f3 f2f2 f3 f0 −f1f3 −f2 f1 f0

⎞⎟⎟⎠ : Ω→M4×4(R).

Thus notations �f ∈ Cp(Ω,M4×1(R)) and Bl(f) ∈ Cp(Ω,M4×4(R)), p ∈ N ∪ {0}might be understood directly.

Denote for a quaternionic constant c,

cΩ : x ∈ Ω→ c ∈ H.

By abuse of notation, we continue to write cΩ for the case of a constant matrix c.

Let ψ := {ψ0, ψ1, ψ2, ψ3} ∈ H4 be a system of quaternions such that theconditions

ψm · ψn + ψn · ψm = 2〈 �ψm, �ψn〉R4 = 2δn,m ∀n,m ∈ N3 ∪ {0}, (1.2)

be fulfilled, where δn,m is the Kronecker symbol and 〈·, ·〉R4 denotes the scalarproduct.

For abbreviation, �ψ := {�ψ0, �ψ1, �ψ2, �ψ3} can be thought of as an orthonormal(in the usual Euclidean sense) basis in R4. In this way we obtain what is knownas structural set, see [13, 18, 19, 20, 24].


Let us now introduce the usual definition of equivalent structural sets.

Definition 1.1. Two structural sets ϕ, ψ are said to be left equivalent (resp. right)if there exists h ∈ H, |h| = 1 such that

ψ = hϕ (resp. ψ = ϕh).

This name is shortened if misunderstanding is excluded.

Remark. Observe that the left equivalence (also the right) represents an equiva-lence relation on the collection of all structural sets. Moreover, each establishedequivalence class has a unique representative structural set of type ψ = {1, ψ1,ψ2, ψ3}.

In geometric terms Definition 1.1 means that there exists a rotation whichmaps the orthonormal basis ϕ into the orthonormal basis ψ.

The following properties of the structural sets of type ψ = {1, ψ1, ψ2, ψ3} areestablished by direct computation.

Proposition 1.2. Let ψ = {1, ψ1, ψ2, ψ3} be a structural set. Then:

i) ψn = −ψn, ∀n ∈ N3,

ii) (ψn)2= −1, ∀n ∈ N3,

iii) ψn · ψm = −ψm · ψn, ∀n,m ∈ N3, n = m,iv) ψ1 · ψ2 · ψ3 = ±1.

Taking into account the non-commutativity of H, every structural set ψ gen-erates Cauchy–Riemann operators (left or right), which are defined in C1(Ω,H)by the following equalities:

ψD[f ] :=

3∑n=0

ψn · ∂xn [f ]; Dψ [f ] :=

3∑n=0

∂xn [f ] · ψn, (1.3)

where ∂xn := ∂/∂xn.

Let ΔH[f ] =∑3

n=0 ΔR4 [fn]en, where ΔR4 =∑3

n=0 ∂2x2n. Then in C2(Ω,H)

the equalities

ψD · ψD = ψD · ψD = Dψ ·Dψ = Dψ ·Dψ = ΔH (1.4)

hold.For fixed ψ and Ω we introduce the set of the so-called ψ-hyperholomorphic

functions (left or right respectively), which are given by

ψM(Ω;H) := ker ψD = {f ∈ C1(Ω;H) : ψD[f ] = 0Ω},Mψ(Ω;H) := kerDψ = {f ∈ C1(Ω;H) : Dψ [f ] = 0Ω}.

In [1] are described all different classes of hyperholomorphic functions, seealso [13, 18, 19, 20, 24].

Theorem 1.3. ψM(Ω,H) = ϕM(Ω,H) if and only if ϕ, ψ are left equivalent.

We emphasize that the set of all classes of hyperholomorphy and the quotientof the collection of structural sets by the left equivalence relation are isomorphic.


2. Necessary and sufficient conditions for associated pairs

Let Ω ⊂ R4 a domain and let f =∑3

n=0 fnen be ψ-hyperholomorphic in Ω. We

check at once that ψD[f ] = 0 is equivalent to ΨD�f = 0, where ΨD is defined onC1(Ω,M4×1(R)) and is given by

ΨD�f :=

3∑n=0

Ψn ∂ �f

∂xn, where Ψn = Bl (ψ

n) and∂ �f

∂xn=

⎛⎜⎜⎜⎜⎝∂f0∂xn

∂f1∂xn

∂f2∂xn

∂f3∂xn

⎞⎟⎟⎟⎟⎠ .

In the same manner we can see the action of ΨD on C1(Ω,M4×4(R)) andconsider the relation between both realization of ΨD.

Let F ∈C1(Ω,M4×4(R)), then F = (f0|f1|f2|f3) where fn ∈ C1(Ω,M4×1(R)),n ∈ N3 ∪ {0} denotes the nth column of F . Using the properties of the matrixproduct the following correlation is valid

ΨDF =(ΨDf0|ΨDf1|ΨDf2|ΨDf3

). (2.5)

Hence, ΨDF = 0 yields the ψ-hyperholomorphicity of fn, n ∈ N3 ∪ {0}.Introduce the first-order differential operator L on C1(Ω,M4×1(R)) as follows,

L�f =

3∑n=0

An∂ �f

∂xn+B�f + C, (2.6)

where An and B areM4×4(R)-valued functions on Ω meanwhile C is one M4×1(R)-valued.

Definition 2.1. A pair of operators ΨD,L is said to be associated if ΨD�f = 0

implies ΨD(L�f)= 0.

We will denote by S1ΨD the set of all operators (2.6) which are associated to ΨD.

Proposition 2.2. Let ψ and ϕ be left equivalent structural sets. Then,

S1ΨD = S1

ΦD. (2.7)

Proof. It suffices to make the following observation

L ∈ S1ΨD ⇔ L

(ψM(Ω,H)

) ⊂ ψM(Ω,H) = ϕM(Ω,H) ⇔ L ∈ S1ΦD. �

Theorem 2.3. Let ψ = {1, ψ1, ψ2, ψ3} be a structural set. Then, the operator (2.6)is associated to ΨD if and only if there exists a first-order linear differential oper-ator L′ (not necessarily associated to ΨD) such that on C2(Ω,M4×1(R)) we have

ΨD L = L′ ΨD. (2.8)


Proof. The sufficiency in this case is clear. Indeed, if there exists L′ satisfying (2.8)

we obtain, for every f such that ΨD�f = 0, that ΨD(L�f)= L′

(ΨD�f

)= 0, by

the linearity of L′.Suppose now that L is associated to ΨD. By (2.6) we have:

L�f = A0ΨD�f +

3∑n=1

(An −A0Ψn)

∂ �f

∂xn+B�f + C.

To shorten notation we write X0 := A0, Xn := An − A0Ψn (n ∈ N3), X4 := B y

X5 := C, then L takes the form

L�f = X0ΨD�f +

3∑n=1

Xn∂ �f

∂xn+X4

�f +X5, (2.9)

where Xn (n=0,1,. . . ,4) are M4×4(R)-valued functions in Ω meanwhile X5 is anM4×1(R)-valued one.

Defining the linear operator L∗ given by

L∗ �f = ΨD(X0

�f)+

3∑n=1

Xn∂ �f

∂xn+X4

�f,

we will calculate the precise expression of RL = ΨD L− L∗ ΨD to make it act onC2(Ω,H).

Let f ∈ C2(Ω,H), then

RL�f = ΨD

(L�f)− L∗

(ΨD�f

)=

3∑n=1

⎡⎣ΨD

(Xn

∂ �f

∂xn

)−Xn

∂(ΨD�f

)∂xn

⎤⎦+[ΨD

(X4

�f)−X4

(ΨD�f

)]+ ΨDX5.

(2.10)

Note that if F = [fij ]i,j=0,...,3 and�f representM4×4(R)- andM4×1(R)-valued

functions respectively, then

∂(F �f)

∂xn=

∂F

∂xn

�f + F∂ �f

∂xn, where

∂F

∂xn=

[∂ fij∂xn

]i,j=0,...,3

.

Thus

ΨD(F �f)=(ΨDF

)�f +

3∑n=0

ΨnF∂ �f

∂xn, and

∂(ΨD�f

)∂xn

=3∑

m=0

Ψm ∂2 �f

∂xn∂xm.


Combining the last two equalities in (2.10) yields

RL�f =

3∑n=1

[(ΨDXn

) ∂ �f

∂xn+

3∑m=0

ΨmXn∂2 �f

∂xm∂xn−

3∑m=0

XnΨm ∂2 �f

∂xn∂xm

]

+(ΨDX4

)�f +

3∑n=0

[ΨnX4

∂ �f

∂xn−X4Ψ

n ∂ �f

∂xn

]+ ΨDX5

=

3∑n,m=1

(ΨmXn −XnΨm)

∂2 �f

∂xm∂xn(2.11)

+

3∑n=1

(ΨDXn +ΨnX4 −X4Ψ

n) ∂ �f

∂xn

+(ΨDX4

)�f + ΨDX5.

Then,

RL�f =

∑1≤n≤m≤3

Anm∂2 �f

∂xm∂xn+

3∑n=1

Bn∂ �f

∂xn+ C �f + D, (2.12)

where,

Anm =

{ΨnXn −XnΨ

n n = m,

(ΨnXm −XmΨn) + (ΨmXn −XnΨm) n = m,

(2.13)

Bn = ΨDXn +ΨnX4 −X4Ψn (2.14)

C = ΨDX4, (2.15)

D = ΨDX5. (2.16)

with n,m ∈ N3.Observe that RL does not depend on ∂x0 . As L is associated to ΨD, the

linearity of L∗ gives RL[f ] = 0Ω for all f ∈ ψM(Ω,H). Having disposed of thispreliminary step, we proceed to show the nullity of RL on C2(Ω,H). To do that,we need to consider the following sequence of assertions:

• When �f = �0Ω is substituted in (2.12) we have

�0Ω = RL�f = D. (2.17)

• Taking in (2.12) ψ-hyperholomorphic functions fk ≡ ek for k ∈ N3 ∪ {0}yields

�0Ω = RL�fk = C �ek k ∈ N3 ∪ {0}.

Then,

C ≡ 0. (2.18)

• For n ∈ N3 let us consider the ψ-hyperholomorphic functions

fkn(x) = (x0 + ψnxn) ek, i.e., �fk

n(x) = x0 �ek +Bl (ψn)xn �ek,

for k ∈ N3 ∪ {0}. Then ∂ �fkn/∂xn = Bl (ψ

n) �ek and,

�0Ω = RL�fkn = BnBl (ψ

n) �ek k ∈ N3 ∪ {0}.The last means that the kth column of BnBl (ψ

n) is �0Ω for k ∈ N3 ∪ {0} andBnBl (ψ

n) ≡ 0⇒ Bn ≡ 0 n ∈ N3. (2.19)

• Now, for n ∈ N3 consider the ψ-hyperholomorphic functions fknn(x) = (x2

0 −x2n + ψn2x0xn)ek for k ∈ N3 ∪ {0}, for which ∂2 �fk

nn/∂x2n = −2 �ek. So we find

�0Ω = RL�fknn = −2Ann �ek, k ∈ N3 ∪ {0}.

One concludes that the kth column of Ann is �0Ω for k ∈ N3∪{0}. This implies

Ann ≡ 0 n ∈ N3. (2.20)

• Finally, for each pair m,n, 1 ≤ n < m ≤ 3 by choosing the ψ-hyperholo-morphic functions fk

nm(x) =(ψn(x2

m − x2n) + ψm2xnxm

)ek for k ∈ N3∪{0},

we get ∂2 �fknm

/∂xn∂xm = 2Bl (ψ

m) �ek and thus

�0Ω = RL�fknm = 2AnmBl (ψ

m) �ek k ∈ N3 ∪ {0}.The result is that the kth column of AnmBl (ψ

m) is �0Ω for k ∈ N3 ∪ {0}obtaining

AnmBl (ψm) ≡ 0⇒ Anm ≡ 0, 1 ≤ n < m ≤ 3. (2.21)

Adding up (2.12), (2.17), (2.18), (2.19), (2.20) and (2.21) we can assert thatRL ≡ 0 on C2(Ω,H). Then, exists L′ = L∗ for which (2.8) is satisfied. �

Next we formulate a criterion under which the operator (2.6) and ΨD forman associated pair.

Theorem 2.4. Let ψ = {1, ψ1, ψ2, ψ3} be a structural set. Then, the operator (2.6)is associated to ΨD if and only if the following conditions are satisfied:

ΨnXm +ΨmXn = XmΨn +XnΨm, , (2.22)

ΨDXn +ΨnB −BΨn = 0, (2.23)

ΨDB = 0, (2.24)

ΨDC = 0, (2.25)

where Xn = An −A0Ψn and n,m ∈ N3.


3. First-order symmetries of the generalizedCauchy–Riemann operator

This section will be devoted to the study of first-order symmetries of the quater-nionic Cauchy–Riemann ψ-operator. In particular, to those symmetries which rep-resent also ϕ-operators of Cauchy–Riemann for some structural set ϕ.

Definition 3.1. A quaternionic first-order partial differential operator L is said tobe a symmetry of ψD if ψD[f ] = 0Ω implies that ψD [L[f ]] = 0Ω.

Taking in (2.6) An = Bl(αn), n ∈ N3 ∪ {0}, B = Bl(β) and C = �γ, whereαn (n ∈ N3 ∪ {0}), β and γ are H-valued functions, we can write L in a quater-nionic form L defined by

L[f ] =

3∑n=0

αn · ∂xn [f ] + β · f + γ. (3.26)

From this transformation and (1.1), Theorem 2.3 shows the following result.

Theorem 3.2. Let ψ = {1, ψ1, ψ2, ψ3} be a structural set. Then, the operator (3.26)is a symmetry of ψD if and only if there exists a quaternionic first-order linearpartial differential operator L′ (not necessarily a symmetry of ψD) such that onC2(Ω,H) it holds

ψD L = L′ ψD. (3.27)

We can now proceed analogously to the proof of the following criterion:

Theorem 3.3. Let ψ = {1, ψ1, ψ2, ψ3} be a structural set. Then, the operator (3.26)is a symmetry of ψD if and only if the following identities hold:

ψn · Xm + ψm · Xn = Xm · ψn + Xn · ψm, (3.28)

ψD[Xn] + ψn · β − β · ψn = 0Ω, (3.29)

ψD[β] = 0Ω, (3.30)

ψD[γ] = 0Ω, (3.31)

where Xn = αn − α0 · ψn and m,n ∈ N3.

Remark. Likewise, we can see that for x ∈ H we have Bl(x) =(�x∣∣∣ �xe1∣∣∣ �xe2∣∣∣ �xe3).

Then, (2.5) enables us to write for f ∈ C1(Ω,H)

ΨD(Bl(f)

)=(ΨD

(�f) ∣∣∣ΨD( �fe1

) ∣∣∣ΨD( �fe2

) ∣∣∣ΨD( �fe3

))=

( −→ψD[f ]

∣∣∣ −→ψD[f ]e1

∣∣∣ −→ψD[f ]e2

∣∣∣ −→ψD[f ]e3

)= Bl

(ψD[f ]

).

This gives (3.29) and (3.30) as direct consequences of (2.23) and (2.24) respectively.


3.1. ψ-symmetries given by ϕ-operators

Let ψ = {1, ψ1, ψ2, ψ3} be a structural set. Our next goal is to determine all thestructural sets ϕ := {ϕ0, ϕ1, ϕ2, ϕ3} such that ϕD be a symmetry of ψD.

According to Theorem 3.3 for L = ϕD, we have that ϕD is a symmetry ofψD if and only if

ψn · Xm + ψm · Xn = Xm · ψn + Xn · ψm, ∀n,m ∈ N3, (3.32)

where Xn =(ϕn − ϕ0 · ψn

)Ω, n ∈ N3 ∪ {0}. Observe that in this case γ = β = 0Ω

and Xn are constant functions in Ω. Then, for all ψD, the conditions (3.29), (3.30)and (3.31) hold. From this, our next concern will be only (3.32).

Writing ϕn = hn · ψn, n ∈ N3 ∪ {0}, (|hn| = 1) yields

Xn = ϕn − ϕ0 · ψn = (hn − h0) · ψn, ∀n ∈ N3. (3.33)

Substituting (3.33) into (3.32) and assuming first n = m and later on n = m,n,m ∈ N3, we deduce that

ψn · (hn − h0) · ψn = (hn − h0) · ψn · ψn (3.34)

ψn · (hm − h0) · ψm + ψm · (hn − h0) · ψn = (hm − h0) · ψm · ψn

+ (hn − h0) · ψn · ψm. (3.35)

Applying Proposition 1.2 to (3.34)–(3.35) we obtain that ϕD is a symmetryof ψD if and only if

ψn · (hn − h0) = (hn − h0) · ψn, (3.36)

ψm · ψn · (hm − hn) = −(hm − hn) · ψm · ψn, (3.37)

for all m,n ∈ N3 with n = m.

According to iv) in Proposition 1.2 we need to consider two cases, but wewill give the proof for ψ1 ·ψ2 ·ψ3 = −1, the other case is left to the reader becauseit is completely analogous and shares the majority of the results. The structuralset ψ under study has the multiplication properties.

ψ1 · ψ2 = −ψ2 · ψ1 = ψ3; ψ2 · ψ3 = −ψ3 · ψ2 = ψ1; ψ3 · ψ1 = −ψ1 · ψ3 = ψ2.

These allow us to write (3.37) in the form:

ψk · (hm − hn) = −(hm − hn) · ψk m,n, k ∈ N3, m = n, m = k, n = k. (3.38)

We next turn to finding the elements of H which simultaneously commute andanti-commute with ψn, n ∈ N3. For this to happen, it is enough to examine onlythe case ψ1, the other cases are left to the reader.

It is easy to check if a = a0 + a1ψ1 + a2ψ

2 + a3ψ3 that,

a · ψ1 = ψ1 · a⇔ a = a0 + a1ψ1, and a · ψ1 = −ψ1 · a⇔ a = a2ψ

2 + a3ψ3.

This established the following characterization


Proposition 3.4. Let ψ = {1, ψ1, ψ2, ψ3} be a structural set and let a = a0+a1ψ1+

a2ψ2 + a3ψ

3, with an ∈ R, n ∈ N3 ∪ {0}. Then, for all each m ∈ N3 we have

·ψm = ψm · a ⇔ a = a0 + amψm and a · ψm = −ψm · a ⇔ a0 + amψm = 0.

By the above results and (3.36) we have hn − h0 = ln + knψn, where n ∈ N3

and ln, kn ∈ R. On the other hand, (3.38) implies for each pair m,n ∈ N3 (m = m)that

hm − hn = (lm − ln) + kmψm − knψn = kmψm − knψ

n ⇒ lm = ln.

Taking k = l1 = l2 = l3 we can write:

hn = (k + knψn) + h0, ∀n ∈ N3. (3.39)

We have thus proved that relation (3.39) between hn, n ∈ N3∪{0} is necessaryand sufficient to give (3.36) y (3.37). Let us now indicate the real numbers k, k1, k2and k3 can be taken for ϕ := {h0, h1 · ψ1, h2 · ψ2, h3 · ψ3} to be a structural set.

Set a unitary quaternion h0 = q0 + q1ψ1 + q2ψ

2 + q3ψ3, i.e., 1 = |h0|2 =

q20 + q21 + q22 + q23 . Then, ϕn · ϕ0 + ϕ0 · ϕn = 0 for n ∈ N3, or equivalently,

0 = hn ·ψn ·h0+h0 ·hn · ψn = hn ·ψn ·h0−h0 ·ψn ·hn ⇔ hn ·ψn ·h0 = h0 ·ψn ·hn.

Replacing (3.39) above

(k + knψn) · ψn · h0 = h0 · ψn · (k − knψ

n)

⇔ (−kn + kψn) · h0 = h0 · (kn + kψn) ,

or equivalently k(ψn · h0 − h0 · ψn

)= kn

(h0 + h0

)= 2knq0.

It follows easily that ψn · h0 − h0 · ψn = 2qn, n ∈ N3. Then

ϕn · ϕ0 + ϕ0 · ϕn = 0 ⇔ kqn = knq0, ∀n ∈ N3. (3.40)

Assuming q0 = 0, we can assert that k = λq0, hence that kn = λqn forn ∈ N3, and finally that

hn = λ (q0 + qn · ψn) + h0, ∀n ∈ N3. (3.41)

But on account of |hn| = 1 we have(λ2 + 2λ

)q20+

(λ2 + 2λ

)q2n = 0. Therefore(

λ2 + 2λ) (

q20 + q2n)= 0⇒ λ = 0 or λ = −2,

since q0 = 0 so q20 + q2n > 0.If λ = 0, then hn = h0 for n ∈ N3, which implies that ϕ is a structural set

equivalent to ψ.We next show that for λ = −2 ϕ become a structural set. It is sufficient to

prove that

0 = ϕn · ϕm + ϕm · ϕn = 2

⟨ −→hn · ψn,

−→hm · ψm

⟩R4

∀n,m ∈ N3, n = m, (3.42)

because the cases n = m ∈ N3 ∪ {0} and m = 0, n ∈ N3 have been alreadyconsidered.


If this is so, we have⎧⎪⎨⎪⎩h1 · ψ1 = (−q0 − q1ψ

1 + q2ψ2 + q3ψ

3) · ψ1 = q1 − q0ψ1 + q3ψ

2 − q2ψ3,

h2 · ψ2 = (−q0 + q1ψ1 − q2ψ

2 + q3ψ3) · ψ2 = q2 − q3ψ

1 − q0ψ2 + q1ψ

3,

h3 · ψ3 = (−q0 + q1ψ1 + q2ψ

2 − q3ψ3) · ψ3 = q3 + q2ψ

1 − q1ψ2 − q0ψ

3,

(3.43)which gives for each pair m,n ∈ N3 (m = n)⟨ −→

hm · ψm,−→

hn · ψn

⟩R4

= 0

and (3.42) is proved. Conversely, suppose that q0 = 0, then (3.40) yields k = 0.From this hn = knψ

n+h0. But |hn| = 1⇒ k2n+2knqn = 0⇒ kn = 0 or kn = −2qn,n ∈ N3.

If for every n ∈ N3 kn = 0 or kn = −2qn, the analysis is reduced to the casesappeared when q0 = 0 and ϕ is a structural set as claimed.

Assume that k1, k2, k3 are not all zero. As before, conditions (3.42) are theonly needed to be proved in order that ϕ became a structural set. The proof fallsnaturally into two parts.

Only one kn assume a non-zero value, e.g., k1 = 0. Then

k1 = 0, k2 = −2q2 = 0, k3 = −2q3 = 0,

and ⎧⎪⎨⎪⎩h1 · ψ1 = (q1ψ

1 + q2ψ2 + q3ψ

3) · ψ1 = −q1 + q3ψ2 − q2ψ

3,

h2 · ψ2 = (q1ψ1 − q2ψ

2 + q3ψ3) · ψ2 = q2 − q3ψ

1 + q1ψ3,

h3 · ψ3 = (q1ψ1 + q2ψ

2 − q3ψ3) · ψ3 = q3 + q2ψ

1 − q1ψ2.

(3.44)

Combining these equalities with conditions (3.42) yields⟨ −→h1 · ψ1,

−→h2 · ψ2

⟩R4

= −2q1q2,⟨ −→h2 · ψ2,

−→h3 · ψ3

⟩R4

= 0,⟨ −→h3 · ψ3,

−→h1 · ψ1

⟩R4

= −2q1q3.

Then, ϕ should be an structural set only if 0 = q1q2 = q1q3 ⇔ q1 = 0, since theassumption q2 = 0 and q3 = 0. Hence, in this case k1 = −2q1 = 0, to concludekn = −2qn for n ∈ N3, which is one of the already analyzed cases.

On the other hand, if in the set {k1, k2, k3} two zero elements appeared, e.g.,k1 and k2, then

k1 = 0, k2 = 0, k3 = −2q3 = 0,

and so⎧⎪⎨⎪⎩h1 · ψ1 = (q1ψ

1 + q2ψ2 + q3ψ

3) · ψ1 = −q1 + q3ψ2 − q2ψ

3,

h2 · ψ2 = (q1ψ1 + q2ψ

2 + q3ψ3) · ψ2 = −q2 − q3ψ

1 + q1ψ3,

h3 · ψ3 = (q1ψ1 + q2ψ

2 − q3ψ3) · ψ3 = q3 + q2ψ

1 − q1ψ2.

(3.45)


As before, conditions (3.42) and the last equalities give⟨ −→h1 · ψ1,

−→h2 · ψ2

⟩R4

= 0,

⟨ −→h2 · ψ2,

−→h3 · ψ3

⟩R4

= −2q2q3,⟨ −→h3 · ψ3,

−→h1 · ψ1

⟩R4

= −2q1q3.

Then, ϕ should be a structural set when 0 = q2q3 = q1q3 ⇔ 0 = q1 = q2, becausethe assumption q3 = 0. Therefore, in this case we have k1 = −2q1 = 0 andk2 = −2q2 = 0, and kn = −2qn for n ∈ N3, hence we are in exactly the samesituation as before.

We have actually proved that given a structural set ψ, with ψ0 = 1, the onlystructural set ϕ being ϕD a symmetry of ψD are those left equivalent to ψ or areof the form

ϕ = {h, h1 · ψ1, h2 · ψ2, h3 · ψ3}, (3.46)

where

h1 = −q0 − q1ψ1 + q2ψ

2 + q3ψ3, h2 = −q0 + q1ψ

1 − q2ψ2 + q3ψ

3,

h3 = −q0 + q1ψ1 + q2ψ

2 − q3ψ3,

for some unitary quaternion h = q0 + q1ψ1 + q2ψ

2 + q3ψ3.

Finally, note that⎧⎪⎨⎪⎩h1 · ψ1 =

(−q0 − q1ψ1 + q2ψ

2 + q3ψ3) · ψ1

h2 · ψ2 =(−q0 + q1ψ

1 − q2ψ2 + q3ψ

3) · ψ2

h3 · ψ3 =(−q0 + q1ψ

1 + q2ψ2 − q3ψ

3) · ψ3

which leads to ⎧⎪⎨⎪⎩ψ1(−q0 − q1ψ

1 − q2ψ2 − q3ψ

3)= −ψ1 · h,

ψ2(−q0 − q1ψ

1 − q2ψ2 − q3ψ

3)= −ψ2 · h,

ψ3(−q0 − q1ψ

1 − q2ψ2 − q3ψ

3)= −ψ3 · h.

The structural set ϕ satisfies the relation (3.46) if and only if it is rightequivalent to ψ = {1,−ψ1,−ψ2,−ψ3}. Hence the following theorem has beenproved.

Theorem 3.5. Let ψ = {1, ψ1, ψ2, ψ3} be a structural set. Then, the Cauchy–Riemann operator ϕD should be a symmetry of ψD if and only if ϕ is eitherleft equivalent to ψ or right equivalent to ψ.

Remark. Due to Proposition 3.4 the classes of structural sets involved in the con-clusion of the above theorem are disjoint.


4. Endomorphisms over the quaternions

Let {1, ψ1, ψ2, ψ3} be a structural set. We want to consider the algebra of en-domorphisms End(H), i.e., the algebra of linear maps T : H �→ H. Hereby wefollow [21]. Obviously, End(H) has to be isomorphic to the full matrix algebra of4× 4-matrices. Now, let a ∈ H and we consider the left multiplication operators

ψk : a �→ ψka

as well as the conjugation operators

ψk : a �→ aψk.

Clearly, these operators satisfy the relations

ψkψj + ψjψk = −2δjk, ψkψj + ψjψk = 2δjk, ψjψk = −ψkψj

and generate an (ultra-hyperbolic) space with bilinear form

B(x+ x, y + y) =∑

xjyj − xj yj .

This bilinear form is invariant under O(3, 3) as well as SO(3, 3) respectively. Thevector space can also be decomposed into

E4 × E4

where E4 = span{ψk} and E4 = span{ψk}. Another representation is given bythe so-called Witt basis

fj =1

2(ψj − ψj) f ′

j =1

2(ψj + ψj)

with the decomposition V 4 × V 4′, whereby V 4 = span{fk} and V 4′ = span{f ′k}.

In the last case we can also introduce the primitive idempotent

I = I1, . . . , Im, Ik = −fkf ′k,

for which we have ψkI = ψkI = f ′kI.

For more details we refer to [21].We consider now the algebra End(Π4) ⊗ End(H). Hereby, the algebra of

scalar polynomial operators End(Π4) contains as subalgebra the algebra of scalardifferential operators with polynomial coefficients D(4). The resulting algebraD(4)⊗ End(H) of quaternionic differential operators with polynomial coefficientsis generated by

• the multiplication operators xj : g(x) �→ xjg(x),• the differential operators ∂j : g(x) �→ ∂jg(x),• the left multiplication operators ψk : g(x) �→ ψkg(x),

• as well as the conjugation operators ψk : g(x) �→ g(x)ψk.

We remark that this algebra contains differential operators acting from both sides.Let us take a short look which operators are preserving the action

L(s)g = sg(sxs),

like our operator ψD. That means that

[P (x, ψD), L(s)] = 0

or in other words, that the operator P (x, ψD) is GL(m)-invariant. Since one canassociate to each differential operator P (x, ψD) ∈ D(4)⊗End(H) a symbol P (x, t)determined by

P (x, ψD)e〈x,t〉 = P (x, t)e〈x,t〉

the question is which symbols P (x, t) are SO(m) invariant, i.e.,

SP (SxS, StS)S = P (x, t) ∀S ∈ SO(m).

Since the algebra of SO(m)-invariant scalar-valued polynomial operatorsP (x, t) are generated by |x|2 = xx, |t|2 = tt and 〈x, t〉 = 1

2 (xt + tx) and the

subgroup which leaves x and t invariant is isomorphic to SO(m− 2), i.e., it is thegroup which leaves

span{x, t} = span{ψ1, ψ2}invariant, we get (cf. [21]) the following theorem.

Theorem 4.1. The algebra of SO(m)-invariant quaternionic differential operatorwith polynomial coefficients is generated by the left and right vector variable oper-

ator ψx· and ·xψ, the left and right Cauchy–Riemann operator ψD and Dψ.

For more details we refer to [21] and, of course, the book [4].

Now, let us take a look at the commutator of ψD with the generators of ouralgebra. Here, simple calculations give us

[ψD, ∂k] = 0, [ψD,Xk] = ψk, [ψD,ψk] = −2∑l �=0,k

ψl∂l,

[ψD, ψk] = −1

2ψk

( 3∑l=1

(ψlf ′0 + ψ0f ′

l )∂l

)= −1

2ψk

( 3∑l=1

(2ψl + ψlψ0 + ψ0ψl)∂l

).

In the same way as in the previous sections we can get as conditions for [ψD,L] =MψD with L =

∑al∂l + b and M =

∑cl∂l + d:∑

k<l

(ψkal − alψk + ψlak − akψ

l)∂k∂l =∑k<l

(clψk + ckψ

l)∂k∂l

ψDal + ψkb− bψk = dψk, ψDb = 0

The principal identities we have is {ψD, ψx} = ψDψx + ψxψD = 2E + 4 as

well as [ψD,E] = ψD, where E =∑3

k=0 xk∂k is the Euler operator. From this andthe above relations we obtain as generators of the Lie algebra of symmetries

∂k (4.47)

E + 2 (4.48)


xk∂l − ∂lxk − 1

2ψkψl (4.49)

2xkE − xx∂k + 5xk − xψk. (4.50)

Corollary 4.2. It is easy to see that the above generators are in fact the infinitesimalgenerators of the conformal group.

Acknowledgment

This work was supported by Portuguese funds through the CIDMA – Center forResearch and Development in Mathematics and Applications, and the PortugueseFoundation for Science and Technology (FCT-Fundacao para a Ciencia e a Tec-nologia), within project PEst-OE/MAT/UI4106/2014.

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[24] N.L. Vasilevsky; M.V. Shapiro, Some questions of hypercomplex analysis. Complexanalysis and applications ’87 (Varna, 1987), 523–531, Publ. House Bulgar. Acad.Sci., Sofia, 1998.

Ricardo Abreu BlayaFacultad de Informatica y MatematicaUniversidad de HolguınHolguın, 80100, Cuba

e-mail: [email protected]

Juan Bory Reyes and Alı Guzman AdanDepartamento de MatematicaUniversidad de OrienteSantiago de Cuba 90500, Cuba

e-mail: [email protected]@csd.uo.edu.cu

Uwe KahlerDepartamento de MatematicaUniversidade de AveiroP-3810-159 Aveiro, Portugal


mailto:[email protected]





Trends in Mathematics, 19–41c© 2014 Springer International Publishing

Generalized Quaternionic Schur Functionsin the Ball and Half-space andKrein–Langer Factorization

Daniel Alpay, Fabrizio Colombo and Irene Sabadini

Abstract. In this paper we prove a new version of Krein–Langer factorizationtheorem in the slice hyperholomorphic setting which is more general thanthe one proved in [9]. We treat both the case of functions with κ negativesquares defined on subsets of the quaternionic unit ball or on subsets of thehalf-space of quaternions with positive real part. A crucial tool in the proofof our results is the Schauder–Tychonoff theorem and an invariant subspacetheorem for contractions in a Pontryagin space.

Mathematics Subject Classification (2010). 47A48, 30G35, 30D50.

Keywords. Schur functions, realization, reproducing kernels, slice hyperholo-morphic functions, Blaschke products.

1. Introduction

1.1. Some history

Functions analytic and contractive in the open unit disk D play an importantrole in various fields of mathematics, in electrical engineering and digital signalprocessing. They bear various names, and in particular are called Schur functions.We refer to [33] for reprints of some of the original works. Andre Bloch’s 1926memoir [20] contains also valuable historical background.

Schur functions admit a number of generalizations, within function theoryof one complex variable and outside. A Cr×s-valued function S analytic in D is a

D. Alpay thanks the Earl Katz family for endowing the chair which supported his research, and

the Binational Science Foundation Grant number 2010117. I. Sabadini was partially supportedby the FIRB Project Geometria Differenziale Complessa e Dinamica Olomorfa.

Switzerland

20 D. Alpay, F. Colombo and I. Sabadini

Schur function if and only if the kernel

KS(z, w) =Ir − S(z)S(w)∗

1− zw

is positive definite in D. In fact, much more is true. It is enough to assume thatthe kernel KS(z, w) is positive definite in some subset Ω of D to insure that S isthe restriction to Ω of a (not necessarily unique) function analytic and contractivein D; see [3, 29].

A Schur function has no poles inside the open unit disk. Motivated by loweringa lower bound given by Caratheodory and Fejer in an interpolation problem, Takagiconsidered in [46, 47] rational functions bounded by 1 in modulus on the unit circleand with poles inside D. These are the first instances of what is called a generalizedSchur function. Later studies of such functions include Chamfy [23], Dufresnoy [32](these authors being motivated by the study of Pisot numbers), Delsarte, Geninand Kamp [26, 28] and Krein and Langer [36, 37, 38], to mention a few names.The precise definition of a generalized Schur function was given (in the setting ofoperator-valued functions) by Krein and Langer [36]:

Definition 1.1. A Cr×s-valued function analytic in an open subset Ω of the unit discis called a generalized Schur function if the kernelKS has a finite number (say κ) ofnegative squares in Ω, meaning that for every choice of N ∈ N, c1, . . . , cN ∈ Cr andw1, . . . , wN ∈ Ω, the N×N Hermitian matrix with (�, j) entry c∗�KS(w�, wj)cj hasat most κ strictly negative eigenvalues, and exactly κ strictly negative eigenvaluesfor some choice of N, c1, . . . , cN , w1, . . . , wN .

In the setting of matrix-valued functions, the result of Krein and Langerstates that S is a generalized Schur function if and only if it is the restriction toΩ of a function of the form

B0(z)−1S0(z),

where S0 is a Cr×s-valued Schur function and B0 is Cr×r-valued Blaschke productof degree κ. Besides [36], there exist various proofs of this result; see for instance[10, p. 141], [22].

That one cannot remove the analyticity condition in the result of Krein andLanger when κ > 0 is illustrated by the well-known counterexample S(z) = δ0(z),where δ0(z) = 0 if z = 0 and δ0(0) = 1 (see for instance [10, p. 82]). Taking intoaccount that znδ0(z) ≡ 0 for n > 0 we have

1− S(z)S(w)∗

1− zw=

1

1− zw−

∞∑n=0

znδ0(z)wnδ0(w)

=1

1− zw− δ0(z)δ0(w).

The reproducing kernel Hilbert space associated with δ0(z)δ0(w) is Cδ0 and has azero intersection with H2(D), and hence the kernel KS has one negative square.

Generalized Quaternionic Schur Functions 21

1.2. The slice hyperholomorphic case

Schur functions have been extended to numerous settings, and we mention in par-ticular the setting of several complex variables [2, 19], compact Riemann surfaces[16] and hypercomplex functions [13, 14]. Generalized Schur functions do not existnecessarily in all these settings.

In [7] we began a study of Schur analysis in the framework of slice hyperholo-morphic functions. The purpose of this paper is to prove the theorem of Krein andLanger (we considered a particular case in [9]) and we treat both the unit ball andhalf-space cases in the quaternionic setting. To that purpose we need in particularthe following:

(i) The notion of negative squares and of reproducing kernel Pontryagin spacesin the quaternionic setting. This was done in [15].

(ii) The notion of generalized Schur functions and of Blaschke products, see [8].(iii) A result on invariant subspaces of contractions in quaternionic Pontryagin

spaces.(iv) The notion of realization in the slice-hyperholomorphic setting, in particular

when the state space is a one-sided (as opposed to two-sided) Pontryaginspace.

The paper contains 6 sections, besides the Introduction. Section 2 containsa quick survey of the Krein–Langer result in the classical case. Section 3 intro-duces slice hyperholomorphic functions and discusses Blaschke products. Section4 contains some useful results in quaternionic functional analysis, among whichSchauder–Tychonoff theorem. In Section 5 we present generalized Schur functionsand their realizations. Finally, in Section 6 we prove the Krein–Langer factoriza-tion for generalized Schur functions defined in a subset of the unit ball and finally,in Section 7, we state the analogous result in the case of the half-space.

2. A survey of the classical case

The celebrated one-to-one correspondence between positive definite functions andreproducing kernel Hilbert spaces (see [17]) extends to the indefinite case, whenone considers functions with a finite number of negative squares and reproducingkernel Pontryagin spaces; see [11, 43, 44]. We recall the definition of the latter forthe convenience of the reader.

A complex vector space V endowed with a sesquilinear form [·, ·] is called anindefinite inner product space (which we will also denote by the pair (V , [·, ·])).The form [·, ·] defines an orthogonality: two vectors v, w ∈ V are orthogonal if[v, w] = 0, and two linear subspaces V1 and V2 of V are orthogonal if every vectorof V1 is orthogonal to every vector of V2. Orthogonal sums will be denoted by thesymbol [+]. Note that two orthogonal spaces may intersect. We will denote by thesymbol [⊕] a direct orthogonal sum. A complex vector space V is a Krein space if


it can be written (in general in a non-unique way) as

V = V+[⊕]V−, (2.1)

where (V+, [·, ·]) and (V−,−[·, ·]) are Hilbert spaces. When the space V− (or, as in[35], the space V+) is finite dimensional (note that this property does not dependon the decomposition), V is called a Pontryagin space. The space V endowed withthe form

〈h, g〉 = [h+, g+]− [h−, g−],

where h = h++h− and g = g++g− are the decompositions of f, g ∈ V along (2.1),is a Hilbert space. One endows V with the corresponding topology. This topologyis independent of the decomposition (2.1) (the latter is not unique, but it is so inthe definite case).

Let now T be a linear densely defined map from a Pontryagin space (P1, [·, ·]1)into a Pontryagin space (P2, [·, ·]2). Its adjoint is the operator T ∗ with domainDom (T ∗) defined by:

{g ∈ P2 : h �→ [Th, g]2 is continuous} .One then defines by T ∗g the unique element in P1 which satisfies

[Th, g]2 = [h, T ∗g]1.

Such an element exists by the Riesz representation theorem.

The operator T is called a contraction if

[Th, Th]2 ≤ [h, h]1, ∀h ∈ Dom(T ),

while it is said to be a coisometry if TT ∗ = I.

Theorem 2.1. A densely defined contraction between Pontryagin spaces of the sameindex has a unique contractive extension and its adjoint is also a contraction.

We refer to [18, 21, 31] for the theory of Pontryagin and Krein spaces, andof their operators.

With these definitions, we can state the following theorem, which gathers themain properties of generalized Schur functions.

Theorem 2.2. Let S be a Cr×s-valued function analytic in a neighborhood Ω of theorigin. Then the following are equivalent:

(1) The kernel KS(z, w) has a finite number of negative squares in Ω.(2) There is a Pontryagin space P and a coisometric operator matrix(

A BC D

): P ⊕ Cs → P ⊕ Cr

such that

S(z) = D + zC(I − zA)−1B, z ∈ Ω. (2.2)


(3) There exists a Cr×s-valued Schur function S0 and a Cr×r-valued Blaschkeproduct B0 such that

S(z) = B0(z)−1S0(z), z ∈ Ω.

As a corollary we note that S can be extended to a function of bounded typein D, with boundary limits almost everywhere of norm less than or equal 1.

We note the following:

(a) When the pair (C,A) is observable, meaning

∩∞n=0 kerCAn = {0} , (2.3)

the realization (2.2) is unique, up to an isomorphism of Pontryagin spaces.(b) One can take for P the reproducing kernel Pontryagin space P(S) with re-

producing kernel KS. When 0 ∈ Ω we have the backward shift realization

Af = R0f,

Bc = R0Sc,

Cf = f(0),

Dc = S(0)c,

where f ∈ P(S), c ∈ Cs and where R0 denotes the backward shift operator

R0f(z) =

⎧⎨⎩f(z)− f(0)

z, z = 0,

f ′(0), z = 0.

See [10] for more details on this construction, and on the related isometric andunitary realizations.

3. Slice hyperholomorphic functions and Blaschke products

Let H be the real associative algebra of quaternions, where a quaternion p isdenoted by p = x0 + ix1 + jx2 + kx3, xi ∈ R, and the elements {1, i, j, k} satisfythe relations i2 = j2 = k2 = −1, ij = −ji = k, jk = −kj = i, ki = −ik = j.As is customary, p = x0 − ix1 − jx2 − kx3 is called the conjugate of p, the realpart x0 = 1

2 (p+ p) of a quaternion is also denoted by Re(p), while |p|2 = pp. Thesymbol S denotes the 2-sphere of purely imaginary unit quaternions, i.e.,

S = {p = ix1 + jx2 + kx3 | x21 + x2

2 + x23 = 1}.

If I ∈ S then I2 = −1 and any nonreal quaternion p = x0 + ix1 + jx2 + kx3

uniquely determines an element Ip = (ix1 + jx2 + kx3)/|ix1 + jx2 + kx3| ∈ S.(We note that later i, j, k may also denote some indices, but the context will makeclear the use of the notation.)Let CI be the complex plane R + IR passing through 1 and I and let x + Iy bean element on CI . Any p = x+ Iy defines a 2-sphere [p] = {x+ Jy : J ∈ S}.


We now recall the notion of slice hyperholomorphic function:

Definition 3.1. Let Ω ⊆ H be an open set and let f : Ω→ H be a real differentiablefunction. Let I ∈ S and let fI be the restriction of f to the complex plane CI . Wesay that f is a (left) slice hyperholomorphic function in Ω if, for every I ∈ S, fIsatisfies

1

2

(∂

∂x+ I

∂

∂y

)fI(x+ Iy) = 0.

We say that f is a right slice hyperholomorphic function in Ω if, for every I ∈ S,fI satisfies

1

2

(∂

∂xfI(x+ Iy) +

∂

∂yfI(x+ Iy)I

)= 0.

The set of slice hyperholomorphic functions on Ω will be denoted by R(Ω).It is a right linear space on H.Slice hyperholomorphic functions possess good properties when they are definedon the so-called axially symmetric slice domains defined below.

Definition 3.2. Let Ω be a domain in H. We say that Ω is a slice domain (s-domainfor short) if Ω ∩R is non empty and if Ω ∩CI is a domain in CI for all I ∈ S. Wesay that Ω is axially symmetric if, for all q ∈ Ω, the sphere [q] is contained in Ω.

A function f slice hyperholomorphic on an axially symmetric s-domain Ω isdetermined by its restriction to any complex plane CI , see [25, Theorem 4.3.2].

Theorem 3.3 (Structure formula). Let Ω ⊆ H be an axially symmetric s-domain,and let f ∈ R(Ω). Then for any x+ Jy ∈ Ω the following formula holds

f(x+ Jy) =1

2[f(x+ Iy) + f(x− Iy) + JI(f(x− Iy)− f(x+ Iy))] . (3.1)

As a consequence of this result, we have the following definition:

Definition 3.4. Let Ω be an axially symmetric s-domain. Let h : Ω∩CI → H be aholomorphic map. Then it admits a (unique) left slice hyperholomorphic extensionext(h) : Ω→ H defined by:

ext(h)(x + Jy) =1

2[h(x+ Iy) + h(x− Iy) + JI(h(x− Iy)− h(x+ Iy))] . (3.2)

Remark 3.5. Let Ω ⊆ H be an axially symmetric s-domain and let f, g ∈ R(Ω).We can define a suitable product, called the -product, such that the resultingfunction f g is slice hyperholomorphic. We first define a product between therestrictions fI , gI of f , g to Ω∩CI . This product can be extended to the whole Ωusing formula (3.2). Outside the spheres associated with the zeroes of f ∈ R(Ω)we can consider its slice regular inverse f−�. Note also that (f g)−� = g−� f−�

where it is defined. We refer the reader to [25, p. 125–129] for the details on the -product and -inverse.

The -product can be related to the pointwise product as described in thefollowing result, [25, Proposition 4.3.22]:


Proposition 3.6. Let Ω ⊆ H be an axially symmetric s-domain, f, g : Ω → H beslice hyperholomorphic functions. Then

(f g)(p) = f(p)g(f(p)−1pf(p)), (3.3)

for all p ∈ Ω, f(p) = 0, while (f g)(p) = 0 when p ∈ Ω, f(p) = 0.

An immediate consequence is the following:

Corollary 3.7. If (f g)(p) = 0 then either f(p) = 0 or f(p) = 0 and g(f(p)−1pf(p))= 0.

Remark 3.8. Corollary 3.7 applies in particular to polynomials, allowing to recovera well-known result, see [39]: if a polynomial Q(p) factors as

Q(p) = (p− α1) · · · (p− αn), αj+1 = αj , j = 1, . . . , n− 1 (3.4)

then α1 is a root of Q(p) while all the other zeroes αj , j = 2, . . . , n belong to thespheres [αj ], j = 2, . . . , n. The decomposition of the polynomial Q, in general, isnot unique.Note that when αj+1 = αj then Q(p) contains the second degree factor p2 +2Re(αj)p+ |αj |2 and the zero set of Q(p) contains the whole sphere [αj ]. We willsay that [αj ] is a spherical zero of the polynomial Q.Remark 3.9. Assume that Q(p) factors as in (3.4) and assume that αj ∈ [α1] forall j = 2, . . . , n. Then the only root of Q(p) is p = α1, see [40, Lemma 2.2.11], [41,p. 519] the decomposition in linear factors is unique, and α1 is the only root of Q.Assume that [αj ] is a spherical zero. Then, for any aj ∈ [αj ] we have

p2 + 2Re(αj)p+ |αj |2 = (p− aj) (p− aj) = (p− aj) (p− aj)

thus showing that both aj and aj are zeroes of multiplicity 1. So we can saythat the (points of the) sphere [αj ] have multiplicity 1. Thus the multiplicity of aspherical zero [αj ] equals the exponent of p2 + 2Re(αj)p+ |αj |2 in a factorizationof Q(p).

The discussion in the previous remark justifies the following:

Definition 3.10. Let

Q(p) = (p− α1) · · · (p− αn), αj+1 = αj , j = 1, . . . , n− 1.

We say that α1 is a zero of Q of multiplicity 1 if αj ∈ [α1] for j = 2, . . . , n.We say that α1 is a zero of Q of multiplicity n ≥ 2 if αj ∈ [α1] for all j = 2, . . . , n.Assume now that Q(p) contains the factor (p2 + 2Re(αj)p + |αj |2) and [αj ] is azero of Q(p). We say that the multiplicity of the spherical zero [αj ] is mj if mj isthe maximum of the integers m such that (p2 + 2Re(αj)p+ |αj |2)m divides Q(p).

Note that the notion of multiplicity of a spherical zero given in [34] is differentsince, under the same conditions described in Definition 3.10, it would be 2mj.

Remark 3.11. The polynomial Q(p) can be factored as follows, see, e.g., [34, The-orem 2.1]:

Q(p) =r∏

j=1

(p2 + 2Re(αj)p+ |αj |2)mj

⎛⎝�s∏i=1

�ni∏j=1

(p− αij)

⎞⎠ a,

where

�∏denotes the -product of the factors, [αi] = [αj ] for i = j, αij ∈ [ai] for

all j = 1, . . . , ni and [ai] = [a�] for i = �. Note that

deg(Q) =r∑

j=1

2mj +

s∑i=1

ni.

Definition 3.12. Let a ∈ H, |a| < 1. The function

Ba(p) = (1− pa)−� (a− p)a

|a| (3.5)

is called a Blaschke factor at a.

Remark 3.13. Using Proposition 3.6, Ba(p) can be rewritten as

Ba(p) = (1 − pa)−1(a− p)a

|a|where p = (1− pa)−1p(1− pa).

The following result is immediate, see [8]:

Proposition 3.14. Let a ∈ H, |a| < 1. The Blaschke factor Ba is a slice hyperholo-morphic function in B.

As one expects, Ba(p) has only one zero at p = a and analogously to whathappens in the case of the zeroes of a function, the product of two Blaschke factorsof the form Ba(p) Ba(p) gives the Blaschke factor with zeroes at the sphere [a].Thus we give the following definition:

Definition 3.15. Let a ∈ H, |a| < 1. The function

B[a](p) = (1 − 2Re(a)p+ p2|a|2)−1(|a|2 − 2Re(a)p+ p2) (3.6)

is called Blaschke factor at the sphere [a].

Theorem 5.16 in [8] assigns a Blaschke product having zeroes at a given setof points aj with multiplicities nj , j ≥ 1 and at spheres [ci] with multiplicitiesmi, i ≥ 1, where the multiplicities are meant as exponents of the factors (p− aj)and (p2−Re(aj)p+ |aj |2), respectively. In view of Definition 3.10, the polynomial(p− aj)

�nj is not the unique polynomial having a zero at aj with the given multi-

plicity nj, thus the Blaschke product∏�nj

j=1 Baj is not the unique Blaschke producthaving zero at aj with multiplicity nj .We give below a form of Theorem 5.16 in [8] in which we use the notion of multi-plicity in Definition 3.10:

Theorem 3.16. A Blaschke product having zeroes at the set

Z = {(a1, n1), . . . , ([c1],m1), . . .}where aj ∈ B, aj have respective multiplicities nj ≥ 1, aj = 0 for j = 1, 2, . . .,[ai] = [aj ] if i = j, ci ∈ B, the spheres [cj ] have respective multiplicities mj ≥ 1,j = 1, 2, . . ., [ci] = [cj ] if i = j and∑

i,j≥1

(ni(1 − |ai|) + 2mj(1 − |cj |)

)<∞ (3.7)

is of the form ∏i≥1

(B[ci](p))mi

�∏i≥1

�ni∏j=1

(Bαij (p)),

where nj ≥ 1, α11 = a1 and αij are suitable elements in [ai], αi j+1 = αij , forj = 2, 3, . . ..

Proof. The fact that (3.7) ensure the convergence of the product follows from [8,Theorem 5.6]. The zeroes of the pointwise product

∏i≥1(B[ci](p))

mi correspondto the given spheres with their multiplicities. Let us consider the product:

�n1∏i=1

(Bαi1(p)) = Bα11(p) Bα12(p) · · · Bα1n1(p).

As we already observed in the proof of Proposition 5.10 in [8] this product admitsa zero at the point α11 = a1 and it is a zero of multiplicity 1 if n1 = 1; if n1 ≥ 2,the other zeroes are α12, . . . , α1n1 where α1j belong to the sphere [α1j ] = [a1]. Thisfact can be seen directly using formula (3.3). Thus, according to Remark 3.8, a1is a zero of multiplicity n1. Let us now consider r ≥ 2 and

�nr∏j=1

(Bαrj (p)) = Bαr1(p) · · · Bαrnr(p), (3.8)

and set

Br−1(p) :=

�(r−1)∏i≥1

�ni∏j=1

(Bαij (p)).

ThenBr−1(p) Bαr1(p) = Br−1(p)Bαr1(Br−1(p)

−1pBr−1(p))

has a zero at ar if and only if Bαr1(Br−1(ar)−1arBr−1(ar)) = 0, i.e., if and only

if αr1 = Br−1(ar)−1arBr−1(ar). If nr = 1 then ar is a zero of multiplicity 1 while

if nr ≥ 2, all the other zeroes of the product (3.8) belongs to the sphere [ar] thus,by Remark 3.8, the zero ar has multiplicity nr. This completes the proof. �

Remark 3.17. In the case in which one has to construct a Blaschke product havinga zero at ai with multiplicity ni by prescribing the factors (p−ai1) · · · (p−aini),aij ∈ [ai] for all j = 1, . . . , ni, the factors in the Blaschke product must be chosenaccordingly (see the proof of Theorem 3.16).


Proposition 3.18. The -inverse of Ba and B[a] are Ba−1 , B[a−1] respectively.

Proof. It follows from straightforward computations, by verifying that the prod-ucts Ba Ba−1 and B[a] B[a−1] equal 1. �

Definition 3.19. A Blaschke product of the form

B(p) =r∏

i=1

(B[ci](p))mi

�s∏i=1

�ni∏j=1

(Bαij (p)), (3.9)

is said to have degree d =∑r

i=1 2mi +∑s

j=1 nj .

Proposition 3.20. Let B(p) be a Blaschke product as in (3.9). Then dim(H(B)) =degB.

Proof. Let us rewrite B(p) as

B(p) =

r∏i=1

(Bci(p) Bci(p))mi

�s∏i=1

�ni∏j=1

(Bαij (p)) =

�d∏j=1

Bβj (p),

d = degB. Let us first observe that in the case in which the factors Bβj aresuch that no three of the quaternions βj belong to the same sphere, then thestatement follows from the fact that H(B) is the span of (1 − pβj)

−�. Moreover(1 − pβ1)

−�, . . . , (1 − pβd)−� are linearly independent in the Hardy space H2(B),

see [5, Remark 3.1]. So we now assume that d ≥ 3 and at least three among theβj ’s belong the same sphere. We proceed by induction. Assume that d = 3 andβ1, β2, β3 belong to the same sphere. Since

KB(p, q) =∑n

pn(1−B(p)B(q)∗)qn =∑n

pn(1−Bβ1(p)Bβ1(q)∗)qn

+Bβ1(p) ∑n

pn(1−Bβ2(p)Bβ2(q)∗)qn r Bβ1(q)

∗

+Bβ1(p) Bβ2(p) ∑n

pn(1−Bβ3(p)Bβ3(q)∗)qn r Bβ1(q)

∗ r Bβ1(q)∗

we haveH(Bβ) = H(Bβ1) +Bβ1 H(Bβ2) +Bβ1 Bβ2 H(Bβ3). (3.10)

Now note that H(Bβ1) is spanned by f1(p) = (1−pβ1)−�, Bβ1 H(Bβ2) is spanned

by f2(p) = Bβ1(p) (1 − pβ2)−� and, finally, Bβ1 Bβ2 H(Bβ3) is spanned by

f3(p) = Bβ1(p) Bβ2(p) (1− pβ3)−�. By using the reproducing property of f1 we

have [f1, f2] = 0 and [f1, f3] = 0 (here [·, ·] denotes the inner product in H2(B)).Observe that

[f2, f3] = [(1− pβ2)−�, Bβ2(p) (1 − pβ3)

−�] = 0

since the left multiplication by Bβ1(p) is an isometry in H2(B) and by the re-producing property of (1 − pβ2)

−�. So f1, f2, f3 are orthogonal in H2(B) and sothey are linearly independent. We conclude that the sum (3.10) is direct and hasdimension 3. Now assume that the assertion holds when d = n and there in B(p)

are at least three Blaschke factors at points on the same sphere. We show that theassertion holds for d = n+ 1. We generalize the above discussion by considering

(H(Bβ1) +Bβ1 H(Bβ2) + · · ·+Bβ1 · · · Bβn−1 H(Bβn) + · · ·++Bβ1 · · · Bβn H(Bβn+1).

(3.11)

Let us denote, as before, by f1(p) = (1 − pβ1)−� a generator of H(Bβ1) and by

fj(p) = Bβ1 · · · Bβj−1 (1 − pβj)−� a generator of Bβ1 · · · Bβj−1 H(Bβj ),

j = 1, . . . , n+1. By the induction hypothesis, the sum of the first n terms is directand orthogonal so we show that [fj, fn+1] = 0 for j = 1, . . . , n. This follows, asbefore, from the fact that the multiplication by a Blaschke factor is an isometryand by the reproducing property. The statement follows. �

We now introduce the Blaschke factors in the half-space

H+ = {p ∈ H : Re(p) > 0}.Definition 3.21. For a ∈ H+ set

ba(p) = (p+ a)−� (p− a).

The function ba(p) is called Blaschke factor at a in the half-space H+.

Remark 3.22. The function ba(p) is defined outside the sphere [−a] and it has azero at p = a. A Blaschke factor ba is slice hyperholomorphic in H+.

As before, we can also introduce Blaschke factors at spheres:

Definition 3.23. For a ∈ H+ set

b[a](p) = (p2 + 2Re(a)p+ |a|2)−1(p2 − 2Re(a)p+ |a|2).The function ba(p) is called Blaschke factor at the sphere [a] in the half-space H+.

We now state the following result whose proof mimics the lines of the proofof Theorem 3.16 with obvious changes. Note that an analog of Remark 3.17 holdsalso in this case.

Theorem 3.24. A Blaschke product having zeroes at the set

Z = {(a1, n1), . . . , ([c1],m1), . . .}where aj ∈ H+, aj have respective multiplicities nj ≥ 1, [ai] = [aj ] if i = j,ci ∈ H+, the spheres [cj ] have respective multiplicities mj ≥ 1, j = 1, 2, . . .,[ci] = [cj ] if i = j and∑

i,j≥1

(ni(1 − |ai|) + 2mj(1 − |cj |)

)<∞

is given by ∏i≥1

(b[ci](p))mi

�∏i≥1

�ni∏j=1

(bαij (p)),

where α11 = a1 and αij are suitable elements in [ai] for i = 2, 3, . . ..


Let f(p) =∑+∞

n=−∞(p − p0)�nan where an ∈ H. Following the standard

nomenclature and [45] we now give the definition of singularity of a slice regularfunction:

Definition 3.25. A function f has a pole at the point p0 if there exists m ≥ 0 suchthat a−k = 0 for k > m. The minimum of such m is called the order of the pole;If p is not a pole then we call it an essential singularity for f ;f has a removable singularity at p0 if it can be extended in a neighborhood of p0as a slice hyperholomorphic function.

A function f has a pole at p0 if and only if its restriction to a complex planehas a pole. In this framework there can be poles of order 0. To give an example,let I ∈ S; then the function (p+ I)−� = (p2 + 1)−1(p− I) has a pole of order 0 atthe point −I which, however, is not a removable singularity, see [25, p. 55].

Definition 3.26. Let Ω be an axially symmetric s-domain in H. We say that afunction f : Ω→ H is slice hypermeromorphic in Ω if f is slice hyperholomorphicin Ω′ ⊂ Ω such that every point in Ω \ Ω′ is a pole and (Ω \ Ω′) ∩ CI has no limitpoint in Ω ∩ CI for all I ∈ S.

4. Some results from quaternionic functional analysis

The tools from quaternionic functional analysis needed in the present paper areof two kinds. On one hand, we need some results from the theory of quaternionicPontryagin spaces, taken essentially from [15]. On the other hand, we also needthe quaternionic version of the Schauder–Tychonoff theorem in order to prove aninvariant subspace theorem for contractions in Pontryagin spaces. More generallywe note that in our on-going project on Schur analysis in the slice hyperholomor-phic setting we were lead to prove a number of results in quaternionic functionalanalysis not readily available in the literature.

Operator theory in (quaternionic) Pontryagin spaces plays an important rolein (quaternionic) Schur analysis, and we here recall some definitions and resultsneeded in the sequel. We refer to [15] for more information.

Definition 4.1. Let V be a right quaternionic vector space. The map

[·, ·] : V × V −→ H

is called an inner product if it is a (right) sesquilinear form:

[v1c1, v2c2] = c2[v1, v2]c1, ∀v1, v2 ∈ V , and c1, c2 ∈ H,

which is Hermitian in the sense that:

[v, w] = [w, v], ∀v, w ∈ V .


A quaternionic inner product space V is called a Pontryagin space if it canbe written as a direct and orthogonal sum

V = V+[⊕]V−, (4.1)

where (V+, [·, ·]) is a Hilbert space, and (V−,−[·, ·]) is a finite-dimensional Hilbertspace. As in the complex case, the space V endowed with the form

〈h, g〉 = [h+, g+]− [h−, g−], (4.2)

where h = h+ + h− and g = g+ + g− are the decompositions of f, g ∈ V along(4.1), is a Hilbert space and the norms associated with the inner products (4.2)are equivalent, and hence define the same topology. The notions of adjoint andcontraction are defined as in the complex case, and Theorem 2.1 still holds in thequaternionic setting:

Theorem 4.2 ([9, Theorem 7.2]). A densely defined contraction between quater-nionic Pontryagin spaces of the same index has a unique contractive extensionand its adjoint is also a contraction.

A key result used in the proof of the Krein–Langer factorization is the fol-lowing invariant subspace theorem.

Theorem 4.3 ([6, Theorem 4.6]). A contraction in a quaternionic Pontryagin spacehas a unique maximal invariant negative subspace, and it is one-to-one on it.

The arguments there follow the ones given in the complex case in [30], andrequire in particular to prove first a quaternionic version of the Schauder–Tychonofftheorem, and an associated lemma. We recall these for completeness:

Lemma 4.4 ([6, Lemma 4.4]). Let K be a compact convex subset of a locally convexlinear quaternionic space V and let T : K → K be continuous. If K contains atleast two points, then there exists a proper closed convex subset K1 ⊂ K such thatT (K1) ⊆ K1.

Theorem 4.5 (Schauder–Tychonoff [6, Theorem 4.5]). A compact convex subset ofa locally convex quaternionic linear space has the fixed point property.

5. Generalized Schur functions and their realizations

The definition of negative squares makes sense in the quaternionic setting sincean Hermitian quaternionic matrix H is diagonalizable: it can be written as H =UDU∗, where U is unitary and D is unique and with real entries. The numberof strictly negative eigenvalues of H is exactly the number of strictly negativeelements ofD, see [48]. The one-to-one correspondence between reproducing kernelPontryagin spaces and functions with a finite number of negative squares, provedin the classical case by [43, 44], extends to the Pontryagin space setting, see [15].

We first recall a definition. A quaternionic matrix J is called a signaturematrix if it is both self-adjoint and unitary. The index of J is the number of


strictly negative eigenvalues of J , and the latter is well defined because of thespectral theorem for quaternionic matrices. See, e.g., [48].

Definition 5.1. Let Ω be an axially symmetric s-domain contained in the unit ball,let J1 ∈ Hs×s and J2 ∈ Hr×r be two signature matrix of the same index, andlet S be a Hr×s-valued function, slice hyperholomorphic in Ω. Then S is called ageneralized Schur function if the kernel

KS(p, q) =

∞∑n=0

pn(J2 − S(p)J1S(q)∗)qn

has a finite number of negative squares, say κ, in Ω. We set κ = indS and call itthe index of S.

We will denote by Sκ(J1, J2) the family of generalized Schur functions ofindex κ.

Lemma 5.2. In the notation of Definition 5.1, let x0 ∈ Ω ∩ R. Let b(p) = p+x0

1+px0.

Then the function S ◦ b is a generalized Schur function slice hyperholomorphic atthe origin and with the same index as S.

Proof. First of all, we note that (1 + px0)−� = (1+ px0)

−1 since x0 ∈ R, and that(1 + px0)

−1 commute with p + x0 thus the rational function b(p) is well defined.The result then follows from the formula

∞∑n=0

pn(J2 − S(b(p))J1S(b(q))∗)qn = (1− x2

0)

× (1 + px0)−1

( ∞∑n=0

b(p)n(J2 − S(b(p))J1S(b(q))∗)b(q)

n

)(1 + qx0)

−1.

(5.1)

To show the validity of (5.1) we use [6, Proposition 2.22] to compute the left-handside which gives

∞∑n=0

pn(J2 − S(b(p))J1S(b(q))∗)qn = (J2 − S(b(p))J1S(b(q))

∗) (1 − pq)−�, (5.2)

where the -product is the left one and it is computed with respect to p. Theright-hand side of (5.1) can be computed in a similar was and gives

(1− x20)(1 + px0)

−1

( ∞∑n=0

b(p)n(J2 − S(b(p))J1S(b(q))∗)b(q)

n

)(1 + qx0)

−1

= (1− x20)(1 + px0)

−1(J2 − S(b(p))J1S(b(q))∗) (1− b(p)b(q))−�(1 + qx0)

−1.

(5.3)

We now note that

(1−b(p)b(q))−� =

(1− p+ x0

1 + px0

q + x0

1 + qx0

)−�

=1

1− x20

(1+px0)(1−pq)−�(1+ qx0)


and substituting this expression in (5.3), and using the property that

(J2 − S(b(p))J1S(b(q))∗) (1 + px0) = (1 + px0)(J2 − S(b(p))J1S(b(q))

∗)

since x0 ∈ R, we obtain

(1− x20)(1 + px0)

−1(J2 − S(b(p))J1S(b(q))∗)

1

1− x20

(1 + px0)(1 − pq)−�(1 + qx0)(1 + qx0)−1

(1 + px0)−1(1 + px0)(J2 − S(b(p))J1S(b(q))

∗)(1− pq)−�

= (J2 − S(b(p))J1S(b(q))∗)(1 − pq)−�

and the statement follows. �The reproducing kernel Pontryagin space P(S) associated with a generalized

Schur function S, namely the space with reproducing kernel KS , is a right quater-nionic vector space, with functions taking values in a two-sided quaternionic vectorspace. To present the counterpart of (2.2) with P(S) as a state space we first recallthe following result, see [6, Proposition 2.22].

Proposition 5.3. Let A be a bounded linear operator from a right-sided quaternionicHilbert P space into itself, and let C be a bounded linear operator from P intoC, where C is a two-sided quaternionic Hilbert space. The slice hyperholomorphicextension of C(I − xA)−1, 1/x ∈ ρS(A) ∩R, is

(C − pCA)(I − 2Re(p)A+ |p|2A2)−1.

We will use the notation

C (I − pA)−� def.= (C − pCA)(I − 2Re(p)A+ |p|2A2)−1. (5.4)

For the following result see [7, 8]. First two remarks: in the statement, anobservable pair is defined, as in the complex case, by (2.3). Next, we denote byM∗ the adjoint of a quaternionic bounded linear operator from a Pontryagin spaceP1 into a Pontryagin space P2:

[Mp1 , p2]P2 = [p1 , M∗p2]P1 , p1 ∈ P1 and p2 ∈ P2.

Theorem 5.4. Let J1 ∈ Hs×s and J2 ∈ Hr×r be two signature matrices of the sameindex, and let S be slice hyperholomorphic in a neighborhood of the origin. Then,S is in Sκ(J1, J2) if and only if it can written in the form

S(p) = D + pC (IP − pA)−�B, (5.5)

where P is a right quaternionic Pontryagin space of index κ, the pair (C,A) isobservable, and the operator matrix

M =

(A BC D

): P ⊕ Hs −→ P ⊕ Hr (5.6)

satisfies (A BC D

)(IP 00 J1

)(A BC D

)∗=

(IP 00 J2

). (5.7)

The space P can be chosen to be the reproducing kernel Pontryagin spaceP(S) with reproducing kernel KS(p, q). Then the coisometric colligation (5.6) isgiven by:

(Af)(p) =

{p−1(f(p)− f(0)), p = 0,

f1, p = 0,

(Bv)(p) =

{p−1(S(p)− S(0))v, p = 0,

s1v, p = 0,

Cf = f(0),

Dv = S(0)v,

(5.8)

where v ∈ Hs, S(p) =∑∞

n=0 pnsn and f ∈ P with power series f(p) =

∑∞n=0 p

nfnat the origin.

Assume now in the previous theorem that r = s, J1 = J2 = J , and thatdimP(S) is finite. Then, equation (5.7) is an equality in finite-dimensional spaces(or as matrices) and the function S is called J-unitary. The function S is moreoverrational and its McMillan degree, denoted by degS, is the dimension of the spaceP(S) (we refer to [7] for the notion of rational slice-hyperholomorphic functions.Suffices here to say that the restriction of S to the real axis is an Hr×r-valuedrational function of a real variable).

The -factorization S = S1 S2 of S as a -product of two Hr×r-valued J-unitary functions is called minimal if deg S = deg S1 +deg S2. When κ = 0, S is aminimal product of elements of three types, called Blaschke–Potapov factors, andwas first introduced by V. Potapov in [42] in the complex case. We give now aformal definition of the Blaschke–Potapov factors:

Definition 5.5. A Hr×r-valued Blaschke–Potapov factor of the first kind (resp.second kind) is defined as:

Ba(p, P ) = Ir + (Ba(p)− Ir)P

where |a| < 1 (resp. |a| > 1) and J, P ∈ Hr×r, J being a signature matrix, and Pa matrix such that P 2 = P and JP ≥ 0.A Hr×r-valued Blaschke–Potapov factor of the third kind is defined as:

Ir − ku (p+ w0) (p− w0)−�u∗J

where u ∈ Hr is J-neutral (meaning uJu∗ = 0), |w0| = 1 and k > 0.

Remark 5.6. In the setting of circuit theory, Blaschke–Potapov factors of the thirdkind are also called Brune sections, see, e.g., [27], [4].

In the sequel, by Blaschke product we mean the product of Blaschke–Potapovfactors.

When κ > 0 there need not exist minimal factorizations. We refer to [11, 12]for examples in the complex-valued case. On the other hand, still when κ > 0 but


for J = Ir, a special factorization exists, as a -quotient of two Blaschke products.This is a special case of the factorization of Krein–Langer. The following resultplays a key role in the proof of this factorization. It is specific of the case J1 = Isand J2 = Ir, which allows us to use the fact that the adjoint of a contractionbetween quaternionic Pontryagin spaces of the same index is still a contraction.

Proposition 5.7. In the notation of Theorem 5.4, assume J1 = Is and J2 = Ir.Then the operator A is a Pontryagin contraction.

Proof. Equation (5.7) expresses that the operator matrix M (defined by (5.6)) isa coisometry, and in particular a contraction, between Pontryagin spaces of sameindex. Its adjoint is a Pontryagin space contraction (see [15]) and we have(

A BC D

)∗(IP 00 Ir

)(A BC D

)≤(IP 00 Is

).

It follows from this inequality that

A∗A+ C∗C ≤ Is. (5.9)

Since the range of C is inside the Hilbert space Hr we have that A∗ is a contractionfrom P into itself, and so is its adjoint A = (A∗)∗. �

6. The factorization theorem

Below we prove a version of the Krein–Langer factorization theorem in the slice hy-perholomorphic setting which generalizes [9, Theorem 9.2]. The role of the Blaschkefactors Ba in the scalar case is played here by the Blaschke–Potapov factors withJ = I.

Theorem 6.1. Let J1 = Is and J2 = Ir, and let S be a Hr×s-valued generalizedSchur function of index κ. Then there exists a Hr×r-valued Blaschke product B0

of degree κ and a Hr×s-valued Schur function S0 such that

S(p) = (B−�0 S0)(p).

Proof. We proceed in a number of steps:

Step 1: One can assume that S is slice hyperholomorphic at the origin.

To check this, we note that whenever f = g h, we have f ◦ b = (g ◦ b) (h◦ b)where b(p) = p+x0

1+px0, x0 ∈ R. This equality is true on Ω∩R+, and extends to Ω by

slice hyperholomorphic extension. Thus, taking into account Lemma 5.2, we nowassume 0 ∈ Ω.

Step 2: Let (5.5) be a coisometric realization of S. Then A has a unique maximalstrictly negative invariant subspace M.

Indeed, A is a contraction as proved in Proposition 5.7. The result thenfollows from Theorem 4.3.

The rest of the proof is as in [9], and is as follows. Let M be the spacedefined in STEP 2, and let AM, CM denote the matrix representations of A and


C, respectively, in a basis of M, and let GM be the corresponding Gram matrix.It follows from (5.9) that

A∗MGMAM ≤ GM − C∗

MCM.

Step 3: The equation

A∗MPMAM = PM − C∗

MCM

has a unique solution. It is strictly negative and M endowed with this metric iscontractively included in P(S).

Recall that the S-spectrum of an operator A is defined as the set of quater-nions p such that A2 − 2Re(p)A + |p|2I is not invertible, see [25]. Then, the firsttwo claims follow from the fact that the S-spectrum of AM, which coincides withthe right spectrum of AM, is outside the closed unit ball. Moreover, the matrixGM − PM satisfies

A∗M(GM − PM)AM ≤ GM − PM,

or equivalently (since A is invertible)

GM − PM ≤ A−∗M (GM − PM)A−1

Mand so, for every n ∈ N,

GM − PM ≤ (A−∗M )n(GM − PM)A−n

M . (6.1)

By the spectral theorem (see [24, Theorem 3.10, p. 616] and [25, Theorem 4.12.6,p. 155] for the spectral radius theorem) we have:

limn→∞ ‖A

−nM ‖1/n = 0,

and so limn→∞ ‖(A−∗M )n(PM −GM)A−n

M ‖ = 0. Thus entry-wise

limn→∞(A−∗

M )n(PM −GM)A−nM = 0

and it follows from (6.1) that GM − PM ≤ 0.

By [9, Proposition 8.8]

M = P(B),

when P is endowed with the PM metric and where B is a rational function with as-sociated de Branges–Rovnyak space which is finite dimensional and an anti Hilbertspace. Such functions B are (inverses of) inner functions, and can be seen, as in [1],to be a finite product of Blaschke–Potapov factors of the second kind. Furthermore:

Step 4: The kernel KS −KB is positive.

Let kM denote the reproducing kernel of M when endowed with the P(S)metric. Then

kM(p, q)−KB(p, q) ≥ 0

and

KS(p, q)− kM(p, q) ≥ 0.


Moreover

KS(p, q)−KB(p, q) = KS(p, q)− kM(p, q) + kM(p, q)−KB(p, q)

and so it is positive definite.

Finally we apply [9, Proposition 5.1] to

KS(p, q)−KB(p, q) = B(p) (Ir − S0(p)S0(q)∗) r B(q)∗

where where S0(p) = B(p) S(p), to conclude that S0 is a Schur function. �

7. The case of the half-space

Since the map (where x0 ∈ R+)

p �→ (p− x0)(p+ x0)−1

sends the right half-space onto the open unit ball, one can translate the previousresults to the case of the half-space H+. In particular the Blaschke–Potapov factorsare of the form

Ba(p, P ) = Ir + (ba(p)− 1)P

where P is a matrix such that P 2 = P and JP ≥ 0 where, in general, J is signaturematrix, and a ∈ H+. The factors of the third type are now functions of the form

Ir − ku (p+ w0)−�u∗J

where u ∈ Hr is such that uJu∗ = 0, and w0 + w0 = 0, k > 0. The various def-initions and considerations on rational J-unitary functions introduced in Section5 have counterparts here. We will not explicit them, but restrict ourselves to thecase J1 = Is and J2 = Ir , and only mention the counterpart of the Krein–Langerfactorization in the half-space setting. We outline the results and leave the proofsto the reader.

In the setting of slice hyperholomorphic functions in H+ the counterpart ofthe kernel

∑∞n=0 p

nqn is

k(p, q) = (p+ q)(|p|2 + 2Re(p)q + q2)−1. (7.1)

Definition 7.1. The Hr×s-valued function S slice hypermeromorphic in an axiallysymmetric s-domain Ω which intersects the positive real line belongs to the classSκ(H+) if the kernel

KS(p, q) = Irk(p, q)− S(p) k(p, q) r S(q)∗

has κ negative squares in Ω, where k(p, q) is defined in (7.1).

The following realization theorem has been proved in [6, Theorem 6.2].


Theorem 7.2. Let x0 be a strictly positive real number. A Hr×s-valued function Sslice hyperholomorphic in an axially symmetric s-domain Ω containing x0 is therestriction to Ω of an element of Sκ(H+) if and only if it can be written as

S(p) = H − (p− x0)(G− (p− x0)(p+ x0)

−1GA)

×( |p− x0|2|p+ x0|2A

2 − 2Re

(p− x0

p+ x0

)A+ I

)−1

F,(7.2)

where A is a linear bounded operator in a right-sided quaternionic Pontryagin spaceΠκ of index κ, and, with B = −(I + x0A), the operator matrix(

B FG H

):

(Πk

Hs

)−→

(Πk

Hr

)is co-isometric. In particular S has a unique slice hypermeromorphic extension toH+. Furthermore, when the pair (G,A) is observable, the realization is unique upto a unitary isomorphism of Pontryagin right quaternionic spaces.

By an abuse of notation, we write

S(p) = H − (p− x0)G ((x0 + p)I + (p− x0)B)−�F

rather than (7.2).

In the following statement, the degree of the Blaschke product B0 is thedimension of the associated reproducing kernel Hilbert space with reproducingkernel KB0 .

Theorem 7.3. Let S be a Hr×s-valued function slice hypermeromorphic in an ax-ially symmetric s-domain Ω which intersects the positive real line. Then, S ∈Sκ(H+) if and only if it can be written as S = B−�

0 S0, where B0 is a Hr×r-valued finite Blaschke product of degree κ, and S0 ∈ S0(H+).

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[32] J. Dufresnoy. Le probleme des coefficients pour certaines fonctions meromorphesdans le cercle unite. Ann. Acad. Sci. Fenn. Ser. A. I, no., 250/9:7, 1958.

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Daniel AlpayDepartment of MathematicsBen-Gurion University of the NegevBeer-Sheva 84105 Israel


Fabrizio Colombo and Irene Sabadini(FC) Politecnico di MilanoDipartimento di MatematicaVia E. Bonardi, 9I-20133 Milano, Italy

e-mail: [email protected]@polimi.it






The Fock Space in theSlice Hyperholomorphic Setting

Daniel Alpay, Fabrizio Colombo, Irene Sabadini and Guy Salomon

Abstract. In this paper we introduce and study some basic properties of theFock space (also known as Segal–Bargmann space) in the slice hyperholo-morphic setting. We discuss both the case of slice regular functions overquaternions and the case of slice monogenic functions with values in a Clif-ford algebra. In the specific setting of quaternions, we also introduce the fullFock space. This paper can be seen as the beginning of the study of infinite-dimensional analysis in the quaternionic setting.

Mathematics Subject Classification (2010). MSC: 30G35, 30H20.

Keywords. Fock space, slice hyperholomorphic functions, quaternions, Cliffordalgebras.

1. Introduction

Fock spaces are a very important tool in quantum mechanics, and also in itsquaternionic formulation; see the book of Adler [1] and the paper [31]. Roughlyspeaking, they can be seen as the completion of the direct sum of the symmetricor anti-symmetric, or full tensor powers of a Hilbert space which, from the pointof view of Physics, represents a single particle. There is an alternative descriptionof the Fock spaces in the holomorphic setting which, in this framework, are alsoknown as Segal–Bargmann spaces.

In this note we work first in the setting of slice hyperholomorphic functions,namely either we work with slice regular functions (these are functions defined onsubsets of the quaternions with values in the quaternions) or with slice monogenicfunctions (these functions are defined on the Euclidean space Rn+1 and have valuesin the Clifford algebra Rn), see the book [21].

D. Alpay thanks the Earl Katz family for endowing the chair which supported his research,and the Binational Science Foundation Grant number 2010117. F. Colombo and I. Sabadiniacknowledge the Center for Advanced Studies of the Mathematical Department of the Ben-Gurion University of the Negev for the support and the kind hospitality during the period inwhich part of this paper has been written.

Switzerland

44 D. Alpay, F. Colombo, I. Sabadini and G. Salomon

Slice hyperholomorphic functions have been introduced quite recently butthey have already several applications, for example in Schur analysis and to definesome functional calculi. The application to Schur analysis started with the paper[6] and it is rapidly growing, see for example [3, 4, 5, 7, 8].

The applications to the functional calculus range from the so-called S-func-tional calculus, which works for n-tuples non necessarily commuting operators,to a quaternionic version of the classical Riesz–Dunford functional calculus, see[23]. The literature on slice hyperholomorphic functions and the related functionalcalculi is wide, and we refer the reader to the book [21] and the references therein.

We note that Fock spaces have been treated in the more classical settingof monogenic functions, see for example the book [22]. In the treatment in [22]no tensor products of Hilbert-Clifford modules are involved. In the framework ofslice hyperholomorphic analysis we have already introduced and studied the Hardyspaces (see [7, 3, 4]), and Bergman spaces (see [18, 20, 19]). Here we begin thestudy of the main properties of the quaternionic Fock spaces.

We start by recalling the definition of the Fock space in the classical complexanalysis case (for the origins of the theory see [24]).For n ∈ N let z = (z1, . . . , zn) ∈ Cn where zj = xj + iyj , xj , yj ∈ R (j = 1, . . . , n)and denote by

dμ(z) := π−nΠnj=1dxjdyj

the normalized Lebesgue measure on Cn. The Fock space of holomorphic functionsf defined on Cn is

Fn :=

{f : Cn �→ C such that

∫Cn

|f(z)|2e−|z|2dμ(z) <∞}. (1.1)

The space Fn with the scalar product

〈f, g〉Fn =

∫Cn

f(z)g(z)e−|z|2dμ(z)

becomes a Hilbert space and the norm is

‖f‖2Fn=

∫Cn

|f(z)|2e−|z|2dμ(z), f ∈ Fn.

The space Fn is called boson Fock space and since we will treat this case in thesequel we will refer to it simply as Fock space. One of its most important propertiesis that it is a reproducing kernel Hilbert space. If we denote by 〈·, ·〉Cn the naturalscalar product in Cn defined by 〈u, v〉Cn :=

∑nj=1 ujvj , for every u, v ∈ Cn we

define the function

ψu(z) = e〈u,v〉Cn = e∑n

j=1 ujvj . (1.2)

We have the reproducing property

〈f, ψu〉Fn = f(u), for all f ∈ Fn.

So there are two equivalent characterizations of the Fock space Fn; one geometric,in terms of integrals (see (1.1)), and one analytic, obtained by the reproducing

Fock Space in the Slice Hyperholomorphic Setting 45

kernel property (or, directly from (1.1)): an entire function f(z) =∑

m∈Nn0amzm

of n complex variables z = (z1, . . . , zn) is in Fn if and only if its Taylor coefficientssatisfy ∑

m∈Nn0

m!|am|2 <∞,

where we have used the multi-index notation. A third characterization is of impor-tance, namely (with appropriate identification, and with ◦ denoting the symmetrictensor product)

Fn = ⊕∞k=0(C

n)◦k.

In this paper we will address some aspects of these three characterizations in thequaternionic and Clifford algebras settings.

The paper consists of four sections besides the introduction. In Section 2we give a brief survey of infinite-dimensional analysis. In Section 3 we study thequaternionic Fock space in one quaternionic variable. We then discuss, in Section 4,the full Fock space. In order to define it, we need to study tensor products of quater-nionic two-sided Hilbert spaces. Tensor product of quaternionic vector spaces havebeen treated in the literature at various level, see, e.g., [11], [31, 30]. This sectionin particular opens the way to study a quaternionic infinite-dimensional analysis.The last section considers the case of slice monogenic functions.

2. A brief survey of infinite-dimensional analysis

There are various ways to introduce infinite-dimensional analysis. We mention herefour related approaches:

1. The white noise space and the Bochner–Minlos theorem: The formula

e−t2

2 =1√2π

∫R

e−u2

2 e−itudu (2.1)

is an illustration of Bochner’s theorem. It is well known that there is no suchformula when R is replaced by an infinite-dimensional Hilbert space. On the otherhand, the Bochner–Minlos theorem asserts that there exists a probability measureP on the space S ′ of real tempered distributions such that

e−‖s‖22

2 =

∫S′

ei〈s′,s〉dP (s′). (2.2)

In this expression, s belongs to the space S of real-valued Schwartz function, theduality between S and S ′ is denoted by 〈s′, s〉 and ‖ · ‖2 denotes the L2(R, dx)norm.

The probability space L2(S ′, P ) is called the white noise space, and is denotedby W . Denoting by Qs the map s′ �→ 〈s′, s〉 we see that (2.2) induces an isometry,which we denote Qf , from the Lebesgue space L2(R, dx) into the white noise


space. We now give an important family of orthogonal basis (Hα, α ∈ �) of thewhite noise space, indexed by the set � of sequences (α1, α2, . . .), with entries in

N0 = {0, 1, 2, 3, . . .} ,where αk = 0 for only a finite number of indices k. Let h0, h1, . . . denote the Her-mite polynomials, and let ξ1, ξ2, . . . be an orthonormal basis of L2(R, dx) (typically,the Hermite functions, but other choices are possible). Then

Hα =

∞∏k=1

hαk(Qξk), (2.3)

and, with the multi-index notation

α! = α1!α2! · · · ,we have

‖Hα‖2W = α!. (2.4)

The decomposition of an element f ∈ W along the basis (Hα)α∈� is called thechaos expansion.

2. The Bargmann space in infinitely many variables: When in (1.2), Cn is replaced

by �2(N), we have the function

ψu(z) = e〈u,v〉�2(N) = e∑∞

j=1 ujvj . (2.5)

The map Hα �→ zα is called the Hermite transform, and is unitary from thewhite noise space onto the reproducing kernel Hilbert space with reproducingkernel (2.5).

3. The Fock space: We denote by ◦ the symmetrized tensor product and by

Γ◦(H) = ⊕∞n=0H◦n,

the symmetric Fock space associated to a Hilbert space H. Then, Γ◦(L2(R, dx))can be identified with the white noise space via the Wiener–Ito–Segal transformdefined as follows (see [37, p. 165]):

ξα = ξ◦αi1

i1◦ · · · ◦ ξ◦αim

im∈ H◦n �→ Hα

This is the starting point of our approach to quaternionic infinite-dimensionalanalysis; see Section 4.

4. The free setting. The full Fock space: It is defined by

Γ(H) = ⊕∞n=0H⊗n,

and allows to develop the free analog of the white noise space theory. See [36, 35]for background for the free setting. See [12] for recent applications to the theoryof non commutative stochastic distributions.

We refer in particular to the papers [33, 34, 13, 14] and the books [26, 27,28, 29, 37, 25] for more information on these various aspects.


3. The Fock space in the slice regular case

The algebra of quaternions is indicated by the symbol H. The imaginary unitsin H are denoted by i, j and k, respectively, and an element in H is of the formq = x0 + ix1 + jx2 + kx3, for x� ∈ R. The real part, the imaginary part and themodulus of a quaternion are defined as Re(q) = x0, Im(q) = ix1 + jx2 + kx3,|q|2 = x2

0 + x21 + x2

2 + x23, respectively. The conjugate of the quaternion q = x0 +

ix1 + jx2 + kx3 is defined by q = Re(q) − Im(q) = x0 − ix1 − jx2 − kx3 and itsatisfies

|q|2 = qq = qq.

The unit sphere of purely imaginary quaternions is

S = {q = ix1 + jx2 + kx3 such that x21 + x2

2 + x23 = 1}.

Notice that if I ∈ S, then I2 = −1; for this reason the elements of S are alsocalled imaginary units. Note that S is a two-dimensional sphere in R4. Given anonreal quaternion q = x0 +Im(q) = x0 + I|Im(q)|, I = Im(q)/|Im(q)| ∈ S, we canassociate to it the two-dimensional sphere defined by

[q] = {x0 + IIm(q)| : I ∈ S}.This sphere has center at the real point x0 and radius |Im(q)|. An element in thecomplex plane CI = R+ IR is denoted by x+ Iy.

Definition 3.1 (Slice regular (or slice hyperholomorphic) functions). Let U be anopen set in H and consider a real differentiable function f : U → H. Denote by fIthe restriction of f to the complex plane CI .The function f is (left) slice regular (or (left) slice hyperholomorphic) if, for everyI ∈ S, it satisfies:

∂IfI(x+ Iy) :=1

2

(∂

∂x+ I

∂

∂y

)fI(x+ Iy) = 0,

on U ∩CI . The set of (left) slice regular functions on U will be denoted by R(U).The function f is right slice regular (or right slice hyperholomorphic) if, for everyI ∈ S, it satisfies:

(fI∂I)(x + Iy) :=1

2

(∂

∂xfI(x+ Iy) +

∂

∂yfI(x+ Iy)I

)= 0,

on U ∩ CI .

The class of slice hyperholomorphic quaternionic-valued functions is impor-tant since power series centered at real points are slice hyperholomorphic: if B =B(y0, R) is the open ball centered at the real point y0 and radius R > 0 and iff : B → H is a left slice regular function then f admits the power series expansion

f(q) =+∞∑m=0

(q − y0)m 1

m!

∂mf

∂xm(y0),

converging on B.

A main property of the slice hyperholomorphic functions is the so-calledRepresentation Formula (or Structure Formula). It holds on a particular class ofopen sets which are described below.

Definition 3.2 (Axially symmetric domain). Let U ⊆ H. We say that U is axiallysymmetric if, for all x+ Iy ∈ U , the whole 2-sphere [x+ Iy] is contained in U .

Definition 3.3 (Slice domain). Let U ⊆ H be a domain in H. We say that U is aslice domain (s-domain for short) if U ∩R is non empty and if U ∩CI is a domainin CI for all I ∈ S.

Theorem 3.4 (Representation Formula). Let U be an axially symmetric s-domainU ⊆ H.

Let f be a (left) slice regular function on U . Choose any J ∈ S. Then thefollowing equality holds for all q = x+ yI ∈ U :

f(x+ Iy) =1

2

[f(x+ Jy) + f(x− Jy)

]+ I

1

2

[J [f(x− Jy)− f(x+ Jy)]

]. (3.1)

Remark 3.5. One of the applications of the Representation Formula is the factthat any function defined on an open set ΩI of a complex plane CI which belongsto the kernel of the Cauchy–Riemann operator can be uniquely extended to aslice hyperholomorphic function defined on the axially symmetric completion ofΩI (see [21]).

We now define the Fock space in this framework.

Definition 3.6 (Slice hyperholomorphic quaternionic Fock space). Let I be anyelement in S. Consider the set

F(H) =

{f ∈ R(H) |

∫CI

e−|p|2 |fI(p)|2dσ(x, y) <∞}

where p = x+ Iy, dσ(x, y) := 1πdxdy. We will call F(H) (slice hyperholomorphic)

quaternionic Fock space.

We endow F(H) with the inner product

〈f, g〉 :=∫CI

e−|p|2gI(p)fI(p)dσ(x, y); (3.2)

we will show below that this definition, as well as the definition of Fock space, donot depend on the imaginary unit I ∈ S.The norm induced by the inner product is then

‖f‖2 =∫CI

e−|p|2 |fI(p)|2dσ(x, y).

We have the following result:

Proposition 3.7. The quaternionic Fock space F(H) contains the monomials pn,n ∈ N which form an orthogonal basis.

Proof. Let us choose an imaginary unit I ∈ S and, for n,m ∈ N, compute

〈pn, pm〉 =∫CI

e−|p|2pmpndσ(x, y).

By using polar coordinates, we write p = ρeIθ and we have

〈pn, pm〉 = 1

π

∫ 2π

0

∫ +∞

0

e−ρ2

ρme−ImθρneInθρ dρ dθ

=1

π

∫ 2π

0

∫ +∞

0

e−ρ2

ρm+n+1eI(n−m)θdρ dθ

=1

2π

∫ 2π

0

eI(n−m)θdθ

∫ +∞

0

e−ρ2

ρm+ndρ2.

Since∫ 2π

0 eI(n−m)θdθ vanishes for n = m and equals 2π for n = m, we have〈pn, pm〉 = 0 for n = m. For n = m, standard computations give

〈pn, pn〉 =∫ +∞

0

e−ρ2

ρ2ndρ2 = n!.

Thus the monomials pn belong to F(H) and any two of them are orthogonal. Wenow show that these monomials form a basis for F(H). A function f ∈ F(H) is

entire so it admits series expansion of the form f(p) =∑+∞

m=0 pmam and thus the

monomials pn are generators. To show that they are independent, we show that if〈f, pn〉 = 0 for all n ∈ N then f is identically zero. We have:

〈f, pn〉 =⟨

+∞∑m=0

pmam, pn

⟩

=

∫CI

e−|p|2pn(

+∞∑m=0

pmam

)dσ(x, y)

and so

〈f, pn〉 = limr→+∞

∫|p|<r, p∈CI

e−|p|2pn(

+∞∑m=0

pmam

)dσ(x, y)

= limr→+∞

+∞∑m=0

(∫|p|<r, p∈CI

e−|p|2pnpmdσ(x, y)

)am

= limr→+∞

∫ r

0

ρ2ne−r2dr2an = n!an,

where we used the fact that the series expansion converges uniformly on |p| < r,thus we can exchange the series with the integration where needed. Thus 〈f, pn〉 =0 for all n if and only if an = 0, i.e., f ≡ 0. �

Proposition 3.8. The definition of inner product (3.2) does not depend on theimaginary unit I ∈ S.


Proof. Let f(p) =∑+∞

m=0 pmam, g(p) =

∑+∞m=0 p

mbm ∈ F(H) and let I ∈ S. Wehave

〈f, g〉 =⟨

+∞∑m=0

pmam,

+∞∑m=0

pmbm

⟩

=

∫CI

e−|p|2⎛⎝+∞∑

m=0

pmam

⎞⎠(+∞∑n=0

pnbn

)dσ(x, y)

=

∫CI

e−|p|2(

+∞∑m=0

ampm

)(+∞∑n=0

pnbn

)dσ(x, y)

so that

〈f, g〉 =+∞∑n=0

∫CI

e−|p|2 anpnpnbndσ(x, y)

=

+∞∑n=0

an

(∫CI

e−|p|2 pnpndσ(x, y))bn

=

+∞∑n=0

n!anbn,

which shows that the computation does not depend on the chosen imaginary unit I.�

Let us recall that the slice regular exponential function is defined by

ep :=+∞∑n=0

pn

n!.

We need to generalize the definition of the function ezw =∑+∞

m=0(zw)m

m! , z, w ∈ C

to the slice hyperholomorphic setting.

We first observe that if we set epq =∑+∞

m=0(pq)m

m! then the function epq doesnot have any good property of regularity: it is not slice regular neither in p nor inq (while ezw is holomorphic in both the variables). Let us consider p as a variableand q as a parameter and set:

epq� =

+∞∑n=0

(pq)�n

n!=

+∞∑n=0

pnqn

n!(3.3)

where the -product (see [21]) is computed with respect to the variable p. It isimmediate that epq� is a function left slice regular in p and right regular in q.

Remark 3.9. The definition (3.3) is consistent with the fact that we are lookingfor a slice regular extension of ezw. In fact, we start from the function ezw =

∑+∞n=0

znwn

n! , which is holomorphic in z seen as an element on the complex planeCI ; we then use the Representation Formula to get the extension to H:

ext(ezw) =1

2(1 − IqI)

+∞∑n=0

znwn

n!+

1

2(1 + IqI)

+∞∑n=0

znwn

n!= eqw

and since w is arbitrary, we get the statement.

We now set kq(p) := epq� and we discuss the reproducing property in the Fockspace.

Theorem 3.10. For every f ∈ F(H) we have

〈f, kq〉 = f(q).

Moreover, 〈kq, ks〉 = esq� .

Proof. We have

〈f, kq〉 =∫CI

e−|p|2epq� f(p)dσ(x, y)

=

∫CI

e−|p|2(

+∞∑n=0

qnpn

n!

)(+∞∑m=0

pmam

)dσ(x, y)

=

+∞∑n=0

+∞∑m=0

qn

n!〈pm, pn〉am

=

+∞∑n=0

qnan

= f(q).

Similarly, we have

〈kq, ks〉 =∫CI

e−|p|2eps� epq� dσ(x, y)

=

∫CI

e−|p|2(

+∞∑n=0

snpn

n!

)(+∞∑m=0

pmqm

)dσ(x, y)

=

+∞∑n=0

+∞∑m=0

sn

n!〈pm, pn〉 q

m

m!

=+∞∑n=0

snqn

n!

= esq� . �

Proposition 3.11. A function f(p) =∑+∞

m=0 pmam belongs to F(H) if and only if∑+∞

m=0 |am|2m! <∞.


Proof. Let us use polar coordinates; with computations similar to those in theproof of Proposition 3.8 and using the Parseval identity, we have∫

CI

e−|p|2 |f(p)|2dσ(x, y) = limr→+∞

1

π

∫ r

0

e−ρ2

∫ 2π

0

|f(ρeIθ)|2ρ dθ dρ

= 2 limr→+∞

∫ r

0

e−ρ2

(+∞∑m=0

ρ2m|am|2)ρ dρ

= 2 limr→+∞

+∞∑m=0

∫ r

0

e−ρ2

ρ2m+1|am|2dρ

=

+∞∑m=0

|am|2m!

and the statement follows. �

4. Quaternion full Fock space and symmetric Fock space

Let V be a right vector space over H. Recall that a quaternionic inner product onV is a map 〈·, ·〉 : V × V → H satisfying the same properties of a complex innerproduct, with the exception of the homogeneity requirement which is replaced by

〈uα, vβ〉 = β〈u, v〉α,and that if V is complete with respect to the norm induced by the inner product,it is called a right quaternionic Hilbert space. A similar definition can be given inthe case of a quaternionic vector space on the left or two-sided.

Let H be a two-sided quaternionic Hilbert space. Then one may consider thequaternionic n-fold Hilbert space tensor power H⊗n defined by

H⊗n = H⊗H⊗ · · · ⊗ H (n times),

where all tensor products are over H.

Remark 4.1. A convenient way of constructing H⊗n is inductively. Recall that ifM is a left R-module and N is a right R-module, then a tensor product of themMR⊗RN is an abelian group together with a bilinear map δ : M×N →MR⊗RNwhich is universal in the sense that for any abelian group A and a bilinear mapf : M ×N → A, there is a unique group homomorphism f : MR ⊗ RN → A suchthat f ⊗ δ = f . If furthermore, M is a right S-module and N is a left T -module,then SMR⊗RNT is a (S, T )-bi-module if one defines sz = (s⊗1)z and zt = z(1⊗t)for z ∈ SMR ⊗ RNT . Since it holds that

(RMS ⊗ SNT )⊗ TPU∼= RMS ⊗ (SNT ⊗ TPU ),

one can define inductively the tensor product of M1, . . . ,Mn, where Mi is a(Ri−1, Ri)-bi-module, and obtain a (R0, Rn)-bi-module,

R0M1R1⊗ R1M2R2

⊗ · · · ⊗ Rn−1MnRn.


For more details see [32, pp. 133–135]. One can also define it non-inductively (see[17, p. 264]).

We make the convention H⊗0 = H, and the element 1 ∈ H is called thevacuum vector and denoted by 1. For the case of two Hilbert spaces in the nextproposition, see also [11, equation (3)].

Proposition 4.2. Let 〈·, ·〉 be the inner product of H, and assume that it satisfiesalso the additional property

〈u, λv〉 = 〈λu, v〉.Then, it induces an inner product on H⊗n,

〈u1 ⊗ · · · ⊗ un, v1 ⊗ · · · ⊗ vn〉 = 〈〈· · · 〈〈〈u1, v1〉u2, v2〉u3, v3〉 · · · 〉un, vn〉,with the same additional property.

Proof. The statement clearly holds for n = 1. By induction,

〈u1 ⊗ · · · ⊗ unα, v1 ⊗ · · · ⊗ vnβ〉 = 〈〈u1 ⊗ · · · ⊗ un−1, v1 ⊗ · · · ⊗ vn−1〉unα, vnβ〉= β〈〈u1 ⊗ · · · ⊗ un−1, v1 ⊗ · · · ⊗ vn−1〉un, vn〉α= β〈u1 ⊗ · · · ⊗ un, v1 ⊗ · · · ⊗ vn〉α,

and

〈v1 ⊗ · · · ⊗ vn, u1 ⊗ · · · ⊗ un〉 = 〈〈v1 ⊗ · · · ⊗ vn−1, u1 ⊗ · · · ⊗ un−1〉vn, un〉= 〈un, 〈v1 ⊗ · · · ⊗ vn−1, u1 ⊗ · · · ⊗ un−1〉vn〉= 〈〈v1 ⊗ · · · ⊗ vn−1, u1 ⊗ · · · ⊗ un−1〉un, vn〉= 〈〈u1 ⊗ · · · ⊗ un−1, v1 ⊗ · · · ⊗ vn−1〉un, vn〉= 〈u1 ⊗ · · · ⊗ un, v1 ⊗ · · · ⊗ vn〉.

For the additional property, we obtain

〈u1 ⊗ · · · ⊗ un, λv1 ⊗ · · · ⊗ vn〉 = 〈〈u1 ⊗ · · · ⊗ un−1, λv1 ⊗ · · · ⊗ vn−1〉un, vn〉= 〈〈λu1 ⊗ · · · ⊗ un−1, v1 ⊗ · · · ⊗ vn−1〉un, vn〉= 〈λu1 ⊗ · · · ⊗ un, v1 ⊗ · · · ⊗ vn〉.

Additivity and positivity are obvious. �Definition 4.3. The quaternionic full Fock module over a Hilbert space H is thespace

F(H) = ⊕∞n=0H⊗n,

with the corresponding inner product.

Definition 4.4. Let u ∈ H. The right-linear map Tu : F(H)→ F(H) defined by

Tu(u1 ⊗ · · · ⊗ un) = u⊗ u1 ⊗ · · · ⊗ un,

is called the creation map. The right-linear map T ∗u : F(H)→ F(H) defined by

T ∗u (u0 ⊗ · · · ⊗ un) = 〈u, u0〉u1 ⊗ · · · ⊗ un,

is called the annihilator map.


The following result is the quaternionic counterpart of a classical result:

Proposition 4.5. T ∗u is the adjoint of Tu.

Proof. The statement follows from

〈T ∗u (u0 ⊗ · · · ⊗ un), v1 ⊗ · · · ⊗ vn〉 = 〈〈u, u0〉u1 ⊗ · · · ⊗ un, v1 ⊗ · · · ⊗ vn〉

= 〈〈u0, u〉u1 ⊗ · · · ⊗ un, v1 ⊗ · · · ⊗ vn〉= 〈u0 ⊗ · · · ⊗ un, u⊗ v1 ⊗ · · · ⊗ vn〉= 〈u0 ⊗ · · · ⊗ un, Tu(v1 ⊗ · · · ⊗ vn)〉. �

Remark 4.6. Note that the isometry u �→ Tu is both left-linear and right-linear.

The complex-valued version of the following proposition appears in [15, 16],where the free Brownian motion is defined and studied. The derivative of thefunction X(t) is studied in [10].

Proposition 4.7 (The non-symmetric quaternionic Brownian motion). Let H =L2(R

+, dx), and consider X(t) = T1[0,t]+ T ∗

1[0,t]. Then

〈X(t)1, X(s)1〉 = min{t, s}.In particular X(t) is self-adjoint, and if one considers the expectation

E : B(F(H))→ H

defined by E(T ) = 〈T1,1〉, thenE(X(s)∗X(t)) = min{t, s}.

Proof. More generally, note that

〈(Tu + T ∗u )1, (Tu + T ∗

u )1〉 = 〈Tu1, Tu1〉 = 〈u, u〉.Since 〈1[0,t],1[0,s]〉H = min{t, s}, the result follows. �

The symmetric product ◦ is defined by

u1 ◦ · · · ◦ un =1

n!

∑σ∈Sn

uσ(1) ⊗ · · · ⊗ uσ(n),

and the closed subspace of H⊗n generated by all vectors of this form is called thenth symmetric power of H, and denoted by H◦n.

Proposition 4.8. Let 〈·, ·〉 be the inner product of H, and assume that it satisfiesalso the additional property

〈u, λv〉 = 〈λu, v〉.Then, it induces an inner product on H◦n

〈u1 ◦ · · · ◦ un, v1 ◦ · · · ◦ vn〉=

1

n!2

∑σ,τ∈Sn

〈〈· · · 〈〈〈uσ(1), vτ(1)〉uσ(2), vτ(2)〉uσ(3), vτ(3)〉 · · · 〉uσ(n), vτ(n)〉,

with the same additional property.

Proof. The result follows as in the proof of Proposition 4.2. �

In the classical case (whereH is a Hilbert space over the field R or C), anothernatural inner-product is usually being used, namely the symmetric inner product.It is defined by

〈u1 ◦ · · · ◦ un, v1 ◦ · · · ◦ vn〉 = per (〈ui, vj〉) ,where per(A) is called the permanent of A and has the same definition as a deter-minant, with the exception that the factor sgn(σ) is omitted. An easy computationimplies that when restricted to the n-fold symmetric tensor power H◦n, the sec-ond inner product (i.e., the symmetric inner product) is simply n! times the firstinner product (the one which is defined in Proposition 4.8). This gives rise to thefollowing definition.

Definition 4.9. Let 〈·, ·〉 be the inner product of H, and assume that it satisfiesalso the additional property

〈u, λv〉 = 〈λu, v〉.Then, the symmetric inner product on H◦n is defined by,

〈u1 ◦ · · · ◦ un, v1 ◦ · · · ◦ vn〉=

1

n!

∑σ,τ∈Sn

〈〈· · · 〈〈〈uσ(1), vτ(1)〉uσ(2), vτ(2)〉uσ(3), vτ(3)〉 · · · 〉uσ(n), vτ(n)〉.

We now focus on the special case of the symmetric Fock space F◦(H) where pis a quaternion variable and H = pH. When no confusion can arise, we will simplydenote it by F◦(H). The following result shows the relation with the Fock spaceas introduced in Definition 3.6, see Proposition 3.11.

Proposition 4.10. F◦(H) is the space of all entire functions∞∑n=0

pnan

satisfying∑∞

n=0 |an|2n! <∞, under an identification of p◦n with pn.

Proof. Clearly, any element in the nth level H◦n can be written as p◦na for somea ∈ H, and

〈p◦n, p◦n〉 = 1

n!

∑σ,τ∈Sn

〈〈· · · 〈〈〈p, p〉p, p〉p, p〉 · · · 〉p, p〉 = n! �

5. The slice monogenic case

In this section we recall just the definition and some properties of slice monogenicfunctions and we show how the results obtained in Section 2 can be reformulatedin this case. We work with the real Clifford algebra Rn over n imaginary unitse1, . . . , en satisfying the relations eiej + ejei = −2δij. An element in the Clifford

algebra Rn is of the form∑

A eAxA where A = i1 . . . ir, i� ∈ {1, 2, . . . , n}, i1 <· · · < ir is a multi-index, eA = ei1ei2 . . . eir and e∅ = 1. We set |A| = i1 + · · ·+ irand we call k-vectors the elements of the form

∑A, |A|=k eAxA, if k > 0. In the

Clifford algebra Rn, we can identify some specific elements with the vectors in theEuclidean space Rn+1: an element (x0, x1, . . . , xn) ∈ Rn+1 will be identified withthe element x = x0+x = x0+

∑nj=1 xjej called, in short, paravector. The norm of

x ∈ Rn+1 is defined as |x|2 = x20 + x2

1 + · · ·+ x2n. The real part x0 of x will be also

denoted by Re(x). Using the above identification, a function f : U ⊆ Rn+1 → Rn

is seen as a function f(x) of the paravector x. We will denote by S the (n − 1)-dimensional sphere of unit 1-vectors in Rn, i.e.,

S = {e1x1 + · · ·+ enxn : x21 + · · ·+ x2

n = 1}.Note that to any nonreal paravector x = x0 + e1x1 + · · ·+ enxn we can associatea (n− 1)-dimensional sphere defined as the set, denoted by [x], of elements of theform x0 + I|e1x1 + · · ·+ enxn| when I varies in S.As it is well known, for n ≥ 3 the Clifford algebra Rn contains zero divisors. Thus,in general, the result which hold in the quaternionic setting do not necessarily holdin Clifford algebra. For this reasons, we quickly revise the definitions and resultsgiven for the quaternionic Fock space. We omit the proofs since, as the reader mayeasily check, the proofs given in the quaternionic case are valid also in this setting.We begin by giving the definition of slice monogenic functions (see [21]).

Definition 5.1. Let U ⊆ Rn+1 be an open set and let f : U → Rn be a realdifferentiable function. Let I ∈ S and let fI be the restriction of f to the complexplane CI . We say that f is a (left) slice monogenic function if for every I ∈ S, wehave

1

2

(∂

∂u+ I

∂

∂v

)fI(u+ Iv) = 0,

on U ∩ CI . The set of (left) slice monogenic functions on U will be denoted bySM(U).

The slice monogenic Fock spaces and their properties are as follows.

Definition 5.2 (Slice hyperholomorphic Clifford–Fock space). Let I be any elementin S. Consider the set

F(Rn+1) =

{f ∈ SM(Rn+1) |

∫CI

e−|x|2|fI(x)|2dσ(u, v) <∞}

where x = u + Iv, dσ(u, v) := 1πdudv. We will call F(Rn+1) (slice hyperholomor-

phic) Clifford–Fock space.

We endow F(Rn+1) with the inner product (which does not depend on thechoice of the imaginary unit I ∈ S):

〈f, g〉 :=∫CI

e−|x|2gI(x)fI(x)dσ(u, v). (5.1)

Proposition 5.3. The Clifford–Fock space F(Rn+1) contains the monomials xm,m ∈ N which form an orthogonal basis, where x is a paravector in Rn+1.

Starting from the function ezy =∑+∞

m=0zmym

m! , holomorphic in z that canbe interpreted as an element on a complex plane CI we can extend it to a slicemonogenic function as

ext(ezy) =1

2(1− IxI)

+∞∑m=0

zmym

m!+

1

2(1 + IxI)

+∞∑m=0

zmym

m!= exy�

and since y is arbitrary, we get the function we need. We now consider the functionky(x) := exy� and we have the reproducing property in the Clifford–Fock space.

Theorem 5.4. For every f ∈ F(Rn+1) we have

〈f, kx〉 = f(x).

Moreover, 〈kx, ky〉 = eyx� .

Proposition 5.5. A function f(x) =∑+∞

m=0 xmam, am ∈ Rn for m ∈ N, belongs to

F(Rn+1) if and only if∑+∞

m=0 |am|2m! <∞.

In the case of modules over Rn the full Fock module is still under investiga-tion.

References

[1] S. Adler, Quaternionic Quantum Field Theory, Oxford University Press, 1995.

[2] D. Alpay, The Schur algorithm, reproducing kernel spaces and system theory, Amer-ican Mathematical Society, Providence, RI, 2001. Translated from the 1998 Frenchoriginal by Stephen S. Wilson, Panoramas et Syntheses.

[3] D. Alpay, V. Bolotnikov, F. Colombo, I. Sabadini, Self-mappings of the quaternionicunit ball: multiplier properties, Schwarz–Pick inequality, and Nevanlinna–Pick inter-polation problem, to appear in Indiana Univ. Math. J., arXiv:1308.2658.

[4] D. Alpay, F. Colombo, I. Lewkowicz, I. Sabadini, Realizations of slice hyperholomor-phic generalized contractive and positive functions, preprint 2013. Submitted.

[5] D. Alpay, F. Colombo, I. Sabadini, Inner product spaces and Krein spaces in thequaternionic setting, in Recent advances in inverse scattering, Schur analysis andstochastic processes. A collection of papers dedicated to Lev Sakhnovich, OperatorTheory Advances and Applications. Linear Operators and Linear Systems, 2014.

[6] D. Alpay, F. Colombo, I. Sabadini, Schur functions and their realizations in the slicehyperholomorphic setting, Integral Equations Operator Theory, 72 (2012), 253–289.

[7] D. Alpay, F. Colombo, I. Sabadini, Pontryagin De Branges Rovnyak spaces of slicehyperholomorphic functions, Journal d’Analyse Mathematique vol. 121 (2013), no.1, 87–125.

[8] D. Alpay, F. Colombo, I. Sabadini, Krein–Langer factorization and related topics inthe slice hyperholomorphic setting, J. Geom. Anal., 24 (2014), no. 2, 843–872.


[9] D. Alpay, A. Dijksma, J. Rovnyak, and H. de Snoo, Schur functions, operator col-ligations, and reproducing kernel Pontryagin spaces, volume 96 of Operator theory:Advances and Applications. Birkhauser Verlag, Basel, 1997.

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[11] D. Alpay, H.T. Kaptanoglu, Quaternionic Hilbert spaces and a von Neumann in-equality, Complex Var. Elliptic Equ., 57 (2012), 667–675.

[12] D. Alpay and G. Salomon. Non-commutative stochastic distributions and applica-tions to linear systems theory. Stochastic Process. Appl., 123(6):2303–2322, 2013.

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[17] N. Bourbaki. Algebra I. Chapters 1–3. Translated from the French. Reprint of the1989 English translation. Elements of Mathematics (Berlin). Springer-Verlag, Berlin,1998.

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[21] F. Colombo, I. Sabadini, D.C. Struppa, Noncommutative Functional Calculus. The-ory and Applications of Slice Hyperholomorphic Functions, Progress in Mathematics,Vol. 289, Birkhauser, 2011.

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[33] I.E. Segal. Tensor algebras over Hilbert spaces. I. Trans. Amer. Math. Soc., 81:106–134, 1956.

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[36] D.V. Voiculescu, K.J. Dykema, and A. Nica. Free random variables, volume 1 ofCRM Monograph Series. American Mathematical Society, Providence, RI, 1992. Anoncommutative probability approach to free products with applications to randommatrices, operator algebras and harmonic analysis on free groups.

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Daniel Alpay and Guy SalomonDepartment of MathematicsBen-Gurion University of the NegevBeer-Sheva 84105 Israel

e-mail: [email protected]@math.bgu.ac.il

Fabrizio Colombo and Irene Sabadini(FC) Politecnico di MilanoDipartimento di MatematicaVia E. Bonardi, 9I-20133 Milano, Italy

e-mail: [email protected]@polimi.it







Multi Mq-monogenic Functionin Different Dimension

Eusebio Ariza and Antonio Di Teodoro

Abstract. A metamonogenic of first-order function or simply metamonogenicfunction is a function that satisfies the differential equation (D − λ)u = 0,where D is the Cauchy–Riemann operator and λ can be real or Clifford-valued constant (see [4]). Using this definition we can say that a multi-metamonogenic function u is separately metamonogenic in several variables

x(j) , j = 1, . . . , n with n ≥ 2, if x(j) = (x(j)1 , . . . , x

(j)mj ) runs in the Euclidean

space Rmj and (Dj − λ)u = 0, for each j = 1, . . . , n,, where Dj is the cor-responding Cauchy–Riemann operator in the space Rmj . Using the theory ofalgebras of Clifford type depending on parameters (see [11, 12]), the presentproposal discusses the properties of u in case the dimensions mj are differ-ent from each other for multi Mq-monogenic functions, following the ideasexhibited in [9, 10].

Mathematics Subject Classification (2010). 30A05; 15A66; 30G35.

Keywords. Monogenic function, metamonogenic function, multi-metamono-genic function, multi Mq-monogenic functions, Clifford algebras, Clifford typedepending on parameters.

1. Introduction

In [2] are defined the n-order meta-analytic functions as the solutions of the equa-tion (∂z − λ)nu = 0. Replacing the operator ∂z by D, a new set of functions areintroduced which we will call n-order meta-monogenic functions. See [1, 4]. Inthis sense, we can see the meta-monogenic functions as a generalization of meta-analytic functions. For n = 1, a continuously differentiable Clifford algebra-valuedfunction, u, is said to be meta-monogenic of first order if it satisfies the equation(D − λ)u = 0. See [1, 4, 14]. We will denote the operator D − λ by Dλ.If we replace the Cauchy–Riemann operator, D, by the more general modified q-

Cauchy–Riemann operator, Dq =n∑

i=0

qiei∂i, for qi real-valued functions defined in

Switzerland

62 E. Ariza and A. Di Teodoro

Rn+1, in the meta of first-order operator, we have a more general operator calledMeta-q of first-order operator or simply Mq operator:

Dq,λ := Dq − λ =

n∑i=0

qiei∂i − λ, (1.1)

where λ can be real or Clifford valued. Solutions of the equation Dq,λu = 0 arecalled Mq-monogenic functions. If λ =

∑ni=0 λiei and λi ∈ R for each i = 0, . . . , n,

the operator (1.1) can be rewritten as:

Dq,λ = Dq − λ = (q0∂0 − λ0)e0 + (q1∂1 − λ1)e1 + · · ·+ (qn∂n − λn)en. (1.2)

This kind of operator is useful in physics (quantum mechanics and electro-magnetism) and was studied recently by Kravchenko in [7] in the context of quater-nionic analysis and is called the electromagnetic Dirac operator.

Example. When n = 2 and the qi, i = 0, 1, 2 are constants or real-valued nonzerofunctions, the Mq-monogenic system or Mq system, Dq,λu = 0 for a Clifford-algebra-valued function

u(x0, x1, x2) = u0(x0, x1, x2) + u1(x0, x1, x2)e1

+ u1(x0, x1, x2)e2 + u12(x0, x1, x2)e12

gives

(q0∂0 − λ0)u0 − (q1∂1 − λ1)u1 − (q2∂2 − λ2)u2 = 0,

(q1∂1 − λ1)u0 + (q0∂0 − λ0)u1 + (q2∂2 − λ2)u12 = 0,

(q2∂2 − λ2)u0 + (q0∂0 − λ0)u2 − (q1∂1 − λ1)u12 = 0,

−(q2∂2 − λ2)u1 + (q1∂1 − λ1)u2 + (q0∂0 − λ0)u12 = 0.

The theory of multi-monogenic functions generalizes the theory of holomor-phic functions in several complex variables to the case of monogenic functions. Inthe holomorphic case the desired function is complex valued. In the monogeniccase it is necessary to choose a suitable space in which the values of the functionare running.

Consider the function u depending on n variables

x(j) = (x(j)0 , . . . , x(j)

mj), j = 1, . . . , n.

Thus, u is defined in the (real) Euclidean space

R = Rm1+1 × Rm2+1 × · · · × Rmn+1

whose dimension is equal ton∑

j=1

mj + n.

Multi Mq-monogenic Function 63

A function u is a Multi-Monogenic Function in several variables x(j), for j =1, . . . , n, if

Dju =

(∂x(j)0

+

mj∑i=1

ei∂(j)i

)u = 0, (1.3)

where ∂(j)i = ∂

∂x(j)i

, and Dj is the Cauchy–Riemann operator in Rmj .

Similarly, a function u is Multi Mq-monogenic in several variables x(j), for j =1, . . . , n, if

Dj,q,λu = 0, (1.4)

where

Dj,q,λ := q0∂x(j)0

+

mj∑i=1

qiei∂(j)i − λ,

q =(q0, q1, . . . , qmj

)and qi are real valued functions defined in Rmj and λ can be

real or Clifford valued.

If n = 1 these functions are the Mq-monogenic functions.

Remark 1.1. Since Dj,1,0 = Dj , we can say that the multi Mq-monogenic functionsare a generalization of the multi-monogenic functions in the same way the Mq-monogenic functions are a generalization of the monogenic functions.

When we consider n ≥ 2, in the simplest case we have the same dimensionmj = m + 1 for all the Euclidean spaces Rmj , and so it is possible to assumethat the desired function u has values belonging to the Clifford algebra Am whichis the usual extension of Rm+1. Since the dimension of the Am equals 2m, theMq-monogenic equations consist of 2m real equations for the 2m real-valued com-ponents of u.

If, however, the numbers mj associated to the variables x(j) are different fromeach other, then m+ 1 can be chosen by

m+ 1 ≥ maxj

mj (1.5)

Thus the number 2m of real-valued components of u is greater than the number2mj−1 of real-value components of Mq-monogenic functions in Rmj for those j forwhich m > mj .

In the specific (1.5) case a theory of multi-monogenic functions (Cauchy’sIntegral Formula, Hartog’s extension theorem, Cousin problem and so on) can befound in [6] as an extension of the works [5, 8] to the case of holomorphic functions.

On the other hand in [10] Tutschke and Hung Son discuss a theory of multi-monogenic functions in the case that the dimension 2m of the corresponding alge-bra of Clifford type depending on parameters will be defined by

m+ 1 =

n∑j=1

mj .


Finally in [9] using the theory of algebras of Clifford type depending on parameters,the authors consider the same real part for all of the factor spaces Rm1+1. Thecorresponding algebraic structure of Clifford type belongs to the product

R× Rm1 × Rm2 × · · · × Rmn .

Following the same ideas presented [9, 10], in this article we develop a theory ofmulti Mq-monogenic functions in case the dimensions mj are different from eachother using an algebraic structure of Clifford type depending on parameters.

2. Separately holomorphic and monogenic functions

Definition 2.1. A separately holomorphic function is a continuous complex-valuedfunction which depends holomorphically on a finite number of complex variables.

Definition 2.2. A separately monogenic function takes its values in the 2m-dim-ensional Clifford algebra, whereas it depends o a finite number of independentvariables all of which run in the same m+ 1-dimensional Euclidean space Rm+1.

In case the independent variables of a separately monogenic function run inEuclidean space of different dimensions, then usually one assumes that the valuesbelong to the Clifford algebra with maximal dimension 2m.

This has the disadvantage that the function has too much components for avariable for which m is not maximal.

Example. Consider the function u holomorphic in the (x0, x1)-plane (u must havetwo components, real and imaginary parts) and monogenic in the (y0, y1, y2)-space(u must have four components in R3). Then the desired function u must have atleast four components depending on x = (x0, x1) and y = (y0, y1, y2), that is

u(x, y) = u0(x, y) + u1(x, y)e1 + u2(x, y)e2 + u12(x, y)e12

where e21 = e22 = −1 and e1e2 + e2e1 = 0.

In case u is a (left-)monogenic function in the y-space, the Cauchy–Riemannsystem

Dyu =

2∑i=0

ei∂yiu = 0,

has to be satisfied.

Since at the same time u is holomorphic in the x-plane, if we put ∂i = ∂xi fori = 0, 1, then

Dxu = (∂0 + e1∂1)u = 0.

Splitting up Dxu in its components, one gets:

∂0u0 − ∂1u1 = 0, ∂0u1 + ∂1u0 = 0,

∂0u2 − ∂1u12 = 0, ∂0u12 + ∂1u2 = 0.


In view of this, the function u, having four components, is holomorphic in thex-plane if and only if the two functions

u0 + u1e1 and u2e2 + u12e12

are holomorphic in the x-plane. Moreover, other possible combinations for u areas follows.Considering the operator

∂0 + e2∂1,

then the holomorphy of u is equivalent to the holomorphy of

u0 + u2e2 and u1e1 + u12e12.

Finally, the holomorphy of u, considering the operator

∂0 + e12∂1

is equivalent to the holomorphy of

u0 + u12e12 and u1e1 + u2e2.

This example shows that, if we have the case that the number of real com-ponents is smaller as the maximal values (which determines the dimension ofthe Clifford algebra), then there exist different possibilities for the choice of theCauchy–Riemann operator to an independent variable.

In order to overcome these differences of the influence of the independentvariables, for each independent variable is assigned its own range.

This means that if the variable x = (x0, . . . , xm) is (m+1)-dimensional, then(for fixed other independent variables) the range of the function is 2m-dimensionalClifford algebra defined by Rm+1.

3. Clifford-algebra-valued functions in several variables

If the basis vector of Rm+1 is {e0, . . . , em}, then

D = ∂0 +

m∑i=1

ei∂i

is the uniquely defined Cauchy–Riemann operator which corresponds to the (m+1)-dimensional variable x = (x0, . . . , xm), with ∂i = ∂xi for i = 0, 1, . . . ,m.

Now consider the function u depending on n variables

x(j) = (x(j)0 , . . . , x(j)

mj) j = 1, . . . , n.

Then, u is defined in the (real) Euclidean space

R = Rm1+1 × Rm2+1 × · · · × Rmn+1

with dimension equalsn∑

j=1

mj + n.


Thus we have the following possibilities:

(a) One possibility is to introduce, for each partial space Rmj+1, an own realpart and mj imaginary units.

Let e1, . . . , em1 be the m1 imaginary unit vectors of Rm1 . Analogously,the mj unit vectors of Rmj are denoted by

ei, i = m1 + · · ·+mj−1 + 1, . . . , m1 + · · ·+mj−1 +mj .

In this case the Cauchy–Riemann operator acting in Rmj+1 is giving by

Dj = ∂x(j)0

+

m1+···+mj∑i=m1+···+mj−1+1

ei∂xi (3.1)

where xm1+···+mj−1+1 = x(j)1 , . . . , x

(j)mj = xm1+···+mj−1+mj .

(b) Another simpler possibility is, for instance, to use the same real part forall of the factor spaces Rm1+1. In other words, the corresponding algebraicstructure of Clifford type belongs to the product

R× Rm1 × Rm2 × · · · × Rmn .

In this case, the Cauchy–Riemann operator acting in Rmj+1 is given by

Dj = ∂x0 +

m1+···+mj−1∑i=m1+···+mj−1+1

ei∂xi (3.2)

for j = 1, . . . , n, where ∂0 correspond to the derivative with respect to thecommon real part x0, m0 = 0 and

xm1+···+mj−1+1 = x(j)1 , . . . , xm1+···+mj−1+mj = x(j)

mj.

Remark 3.1. Note that the derivatives with respect to the variables xi wherei = m1,m1 +m2, . . . ,m1 +m2 + · · ·+mn does not appear in the operator Dj .

Note, additionally, that in the operator Dj we have the unit vectors to thecorresponding Clifford algebra Amj . On the other hand, if we consider the oper-ators given by (3.1) and (3.2), the unit vectors are those of the Clifford algebraAm, where m depends on the mj for j = 1, . . . , n. See Sections 4 and 6.

In order to multiply vectors, it is necessary to introduce an algebraic structureof Clifford type.

4. Associated algebra of Clifford type 1

Clifford algebras over Rm+1 can be constructed as equivalence classes in the ringR[X1, . . . , Xm] of polynomials in m variables X1, . . . , Xm with real coefficients,where two polynomials are said to be equivalent if their difference is a polynomialfor which each term contains at least one of the factors

X2j + 1 and XiXj +XjXi , (4.1)


where i, j = 1, . . . ,m and i = j. Denoting Xj by ej , j = 1, . . . ,m, one obtains theusual Clifford algebra Am, which extends the Euclidean space Rm+1 whose basisis e0 = 1, e1, . . . , em. The structure polynomials (4.1) yield the well-known rulese2j = −1 and eiej + ejei = 0 for its basis elements, i = j. See [3].

In 2008 the authors of [11] introduced the Clifford algebras depending on pa-rameters in order to describe more general systems of partial differential equationsand this is possible in the framework of classical Clifford analysis. These algebrascan be obtained if the structure polynomials (4.1) are replaced by

Xkj

j + αj and XiXj +XjXi − 2γij , (4.2)

where i, j = 1, . . . , n, i = j, and the kj ≥ 2 are natural numbers. The parametersαj and γij = γji have to be real and may depend also on further variables such asthe variable x in Rn+1. If the parameters do not depend on further variables andif n ≥ 3, the Clifford type algebra generated by the structure polynomials (4.2) isdenoted by An(kj , αj , γij). For n = 1 we write A1(k, α). If in An(kj , αj , γij) allof the kj are equal to 2 and αj and γij are constant we denote this algebra byA∗

n,2. This algebra has the dimension 2n. We denoted by An the classical Cliffordalgebra An(2, 1, 0).

In this part we will develop a theory of multi Mq-monogenic functions in thecase that the dimension 2m of the corresponding algebra of Clifford type will bedefined by

m+ 1 =

n∑j=1

mj (4.3)

and we will consider Rm+1.

Remark 4.1. The choice of m in this way has the advantage that the dimensionsmj of the n given Euclidean spaces Rmj have the same influence on the choice ofthe dimension m. In other terms, no space Rmj is preferred in comparison withthe other spaces.

Denote the basis vectors of

Rm1 × Rm2 × · · · × Rmn .

by

{e0 = 1, . . . , em1−1; em1 , . . . , em1+m2−1; . . . , em−mn , . . . , em}.Since, in this case (4.3), the uniform use of the n Euclidean spaces Rmj requiresthe introduction of n real axes, the construction of the related algebra has to bemodified. This construction can be realized in the framework of algebras of Cliffordtype depending on parameters. This algebra allows us to repeat the arguments ofcase (1.5) also in case (4.3).

In order to multiply vectors, we introduce the following algebraic structure ofClifford type. Consider the (non-commutative) ring of polynomials in X1, . . . , Xm.


Let j be one of the indices

j = m1,m1 +m2, . . . ,m1 + · · ·+mn−1,

whereas k and l are indices (between 1 and m) which are different from these n−1indices j. Then the algebra Am(σ1) is defined by the structure relations

e2j = 1, e2k = −1,ekej = ejek, ekel = −elek. (4.4)

The Dj,q,λ operator in the space Rmj is given by

Dj,q,λ = Dj,q − λ = q0∂0 +

m1+···+mj−1∑i=m1+···+mj−1+1

qiei∂xi − λ (4.5)

for j = 1, . . . , n, where ∂0 correspond to the derivative with respect to the commonreal part x0, m0 = 0 and

xm1+···+mj−1+1 = x(j)1 , . . . , x(j)

mj= xm1+···+mj−1+mj .

Note that, like for the operator Dj , the derivatives with respect to the vari-ables xi where

i = m1, m1 +m2, . . . , m1 +m2 + · · ·+mn,

does not appear in the operator Dj,q. This corresponds to the derivatives withrespect to

x(1)0 , . . . , x

(n)0 .

A solution of the system Dj,q,λu = 0 is said to be Mq-monogenic in thespace Rmj .

4.1. Decomposition of the q-Cauchy–Riemann system

Let Dj,q the q-Cauchy–Riemann operator (3.2) of the x(j)-space, for j = 1, . . . , n.Since Am has dimension 2m, the q-Cauchy–Riemann system Dj,qu = 0 can bedecomposed into 2m real equations for the 2m real-valued components of u. Onthe other hand, the differential operator Dj,q contains mj differentiations. Note,additionally, that the usual Clifford algebra associated to Rmj has the dimension2mj−1. In consequence, we can decompose Dj,qu = 0 into 2m−mj+1 groups ofCauchy–Riemann equations in Rmj for 2mj−1 desired real-valued components ata time.

5. Example 1

Consider a function u depending on five real variables x0, x1, y0, y1, y2 which isMq-holomorphic in the x = (x0, x1)-plane but at the same time a Mq-monogenicfunction in the y = (y0, y1, y2)-space. Since

m+ 1 = 5 = m1 +m2,

we have m = 4, whereas the corresponding algebra of Clifford type is A4(σ1).


Thus,

R4+1 = R2 × R3,

which implies

j = 2 and k, l ∈ {1, 3, 4}.The structure relations for the five basis elements {e0 = 1, e1, e2, e3, e4} are

given by

e21 = −1, e22 = 1, e23 = −1, e24 = −1,e1e2 = e2e1, e2e3 = e3e2, e2e4 = e4e2,

e1e3 = −e3e1, e1e4 = −e4e1, e3e4 = −e4e3.Since m = 4, the desired function u has 16 real-valued components

u = u0 + u1e1 + u2e2 + u3e3 + u4e4

+ u12e12 + u13e13 + u14e14 + u23e23 + u24e24 + u34e34

+ u123e123 + u124e124 + u134e134 + u234e234 + u1234e1234.

LetDx,q,λ = D1,q,λ = D1,q − λ = (q0∂0 + q1e1∂1)− λ

be the Mq-monogenic operator in the (x0, x1)-plane. The Mq-monogenic equationsDx,q,λu = 0 implies the following 8 equations for the real components of u:

q0∂0u0 − q1∂1u1 − λu0 = 0, q1∂0u1 + q0∂1u0 − λu1 = 0,

q0∂0u2 − q1∂1u12 − λu2 = 0, q0∂0u12 + q1∂1u2 − λu12 = 0,

q0∂0u3 − q1∂1u13 − λu3 = 0, q0∂0u13 + q1∂1u3 − λu13 = 0,

q0∂0u4 − q1∂1u14 − λu4 = 0, q0∂0u14 + q1∂1u4 − λu14 = 0,

q0∂0u23 − q1∂1u123 − λu23 = 0, q0∂0u123 + q1∂1u23 − λu123 = 0,

q0∂0u24 − q1∂1u124 − λu24 = 0, q0∂0u124 + q1∂1u24 − λu124 = 0,

q0∂0u34 − q1∂1u134 − λu34 = 0, q0∂0u134 + q1∂1u34 − λu134 = 0,

q0∂0u234 − q1∂1u1234 − λu234 = 0, q0∂0u1234 + q1∂1u234 − λu1234 = 0.

Therefore the following eight functions turn out to be Mq-holomorphic in thex0, x1-plane

u0 + u1e1, u2 + u12e1, u3 + u13e1, u4 + u14e1,

u23 + u123e1, u24 + u124e1, u34 + u134e1, u234 + u1234e1.

This can also be seen by the following decomposition of the function u:

u = (u0 + u1e1) + (u2 + u12e1)e2 + (u3 + u13e1)e3 + (u23 + u123e1)e2e3

+ (u24 + u124e1)e2e4 + (u34 + u134e1)e3e4 + (u234 + u1234e1)e2e3e4.

Analogously, the Mq-monogenic equation in the (y0, y1, y2)-space is

(D2,q − λ)u = (q0∂0 + q3e3∂y1 + q4e4∂y2 − λ)u = 0.


This shows that the four functions

u0 + u3e3 + u4e4 + u34e34,

u1e1 + u13e13 + u14e14 + u134e134,

u2e2 + u23e23 + u24e24 + u234e234,

u12e12 + u123e123 + u124e124 + u1234e1234

are Mq-monogenic in the (y0, y1, y2)-space.This can be seen by the following decomposition of the function u:

u = u0 + u3e3 + u4e4 + u34e34

+ e1 (u1 + u13e3 + u14e4 + u134e34)

+ e2 (u2 + u23e3 + u24e4 + u234e34)

+ e12 (u12 + u123e3 + u124e4 + u1234e34) .

6. Associated algebra of Clifford type 2

Consider the real Euclidean space R given by

R = Rm1+1 × Rm2+1 × · · · × Rmn+1.

Define m by m = m1 + · · ·+mn and introduce the algebraic structure of Cliffordtype defined by

e2i = −1 for each i = 1, . . . ,m,

ekel = −elek if k and l belong to the same space Rmj ,

ekel = elek if k and l belong to different spaces Rmj .

(6.1)

The structure relations (6.1) define the algebras of Clifford type depending onparameters Am(σ2).

Let Dj,q be the q-Cauchy–Riemann operator acting in Rmj+1 given by (3.1).That is,

Dj,q = q0∂x(j)0

+

m1+···+mj∑i=m1+···+mj−1+1

qiei∂xi

where xm1+···+mj−1+1 = x(j)1 , . . . , x

(j)mj = xm1+···+mj−1+mj .

In this case, the Dj,q,λ operator acting in Rmj+1 is given by Dj,q,λ = Dj,q − λ,where λ can be real or Clifford valued. A solution of the equation Dj,q,λu = 0 issaid to be Mq-monogenic in the Rmj+1 space.

Then we can define a new class of multi Mq-monogenic functions by thesolutions of the equations Dj,q,λu = (Dj,q − λ) = 0, where the function u is aAm-valued function defined in R.

Note that the commutativity of the unit vectors corresponding to differentspaces implies that

Dj1,qDj2,q = Dj2,qDj1,q if j1 = j2. (6.2)


Since the q-Cauchy–Riemann system in the subspace Rmj consists of 2mj equationsand the function u has 2m real components, the q-Cauchy–Riemann system in thesubspace Rmj can be decomposed in 2m−mj subsystems.

7. Example 2

Consider a function Mq-holomorphic in the (x0, x1)-plane and, at the same time,Mq-monogenic in the (y0, y1, y2)-space (with respect to the operatorDj,q,λ = Dj,q−λ). We assume that the values of u belong to an algebraic structure whose basisvectors belong to

R× R1 × R2.

The dimension is equal

m = m1 +m2 = 1 + 2.

The basis is

{e0, e1, e2, e3, e12, e13, e23, e123},thus u has 23 = 8 real components:

u = u0e1 + u2e2 + u3e3 + u12e12 + u13e13 + u23e23 + u123e123.

The corresponding algebra of Clifford type is A3(σ2), thus

R3 = R1 × R2, j = 1, k ∈ {2, 3}.e21 = 1, e22 = −1, e23 = −1,e1e2 = e2e1, e1e3 = e3e1,

e2e3 = −e3e2.The Mq-holomorphic system Dx,q,λu = D1,q,λu = (q0∂0 + q1e1∂1 − λ)u = 0 in thex = (x0, x1)-plane leads to the four Mq-holomorphic functions

u0 + u1e1, u2 + u12e1, u3 + u13e1, u23 + u123e1.

This can also be seen by the following decomposition of the function u:

u = (u0 + u1e1) + (u2 + u12e1)e2 + (u3 + u13e1)e3 + (u23 + u123e1)e2e3.

Analogously, the Mq-monogenic system in the (y0, y1, y2)-space

D2,q,λu = (q0∂y0 + q1e2∂y1 + q2e3∂y2 − λ)u = 0

show that the following two

u0 + u2e2 + u3e3 + u23e23,

u1e1 + u12e12 + u13e13 + u123e123

are Mq-monogenics.This corresponds to the following decomposition of the function u:

u = (u0 + u2e2 + u3e3 + u23e23) + e1(u1 + u12e2 + u13e3 + u123e23).


8. Definition of separately Mq-monogenic functions

Definition 8.1. A separately Mq-monogenic function takes its values in the 2m-dimensional Clifford algebra defined by (Am(σ1)) or (Am(σ2)), whereas it dependson a finite number of independent variables all of with run in the same m + 1-dimensional Euclidean space Rm+1.

Definition 8.2. A Multi Mq-monogenic function u in Ω takes its values in thealgebraic structureAm(σ1) orAm(σ2) of Clifford type where the structure relationsare given by (4.4) (respectively, (6.1)).

Remark 8.3. The above definition of multi Mq-monogenic functions includes thecase that u is separately Mq-holomorphic with respect to one or several variablesx(j). This is the case if mj = 1. In case mj = 1 for each j = 1, . . . , n, then one getsa new class of separately Mq-holomorphic functions in several complex variables.

9. Conclusions

In this article, we use the theory of algebras of Clifford type depending on pa-rameters [11, 12] to construct suitable structure relations so that the separatelyMeta-q of firstorder monogenic function or Mq-monogenic function has the correctnumber of components with respect to each separate variables following the ideasof the papers [9, 10].

As a natural extension of this article, may be considered more general con-cepts of algebras of Clifford type. See [13]. Also the article leads to representa-tions of the type of the Cauchy Integral Formula for the new classes of multi Mq-monogenic functions. Using power-series representations of the new Cauchy kernel,one also obtains power-series representations for multi Mq-monogenic functions inthe new sense.

Acknowledgment

The authors would like to express their sincere gratitude to Professor WolfgangTutschke for his suggestions in this investigation.

References

[1] Balderrama, C., Di Teodoro, A. and Infante, A., Some Integral Representation forMeta-Monogenic Function in Clifford Algebras Depending on Parameters, Adv. Appl.Clifford Algebras, 23, 4, 793–813, (2013).

[2] Balk, M., Polyanalytic functions. Berlin: Akademie Verlag, 1991.

[3] Brackx, F. Delanghe, R and Sommen, F., Clifford Analysis. Pitman Research Notes,1982.

[4] Di Teodoro, A. and Vanegas, C., Fundamental Solutions for the First-order Meta-Monogenic Operator, Adv. Appl. Clifford Algebras, 22, 1, 49–58, (2012).


[5] Hormander, L., An Introduction to Complex Analysis in Several Variables, North-Holland Mathematical Library, 3rd ed., vol. 7, North-Holland Publishing Co., Ams-terdam, 1990.

[6] Hung Son, L., Recent Trends in Theory of Regular Functions Taking Value in Clif-ford Algebra, Proceedings of the International Conference of Applied Mathematics(ICAM), Hanoi, August 25–29, 2004, SAS International Publications, Delhi, pp.113–122, 2004.

[7] Kravchenko, V., Applied Quaternionic Analysis, Heldermann Verlag, 2003.

[8] Shabat, D.V., Introduction to Complex Analysis, Part II, Functions of several vari-ables. (Translated from the third (1985) Russian edition by J.S. Joel), Translationsof Mathematical Monographs, vol. 110. American Mathematical Society. Providence,RI, 1992.

[9] Tutschke, W. and Hung Son, L., A New Concept of Separately Holomorphic andSeparately Monogenic Functions. Algebraic structures in partial differential equa-tions related to complex and Clifford analysis, Ho Chi Minh City Univ. Educ. Press,Ho Chi Minh City, 67–78, 2010.

[10] Tutschke, W. & Hung Son, L., Multi-monogenic functions in different dimensions.Complex Variables and Elliptic Equations: An International Journal, vol. 58, 2, 293–298, 2013.

[11] Tutschke, W. and Vanegas, C.J., Clifford algebras depending on parameters and theirapplications to partial differential equations. Contained in Some topics on value dis-tribution and differentiability in complex and p-adic analysis. Science Press Beijing,430–449, 2008.

[12] Tutschke, W. and Vanegas, C.J., Metodos del analisis complejo en dimensiones su-periores. Ediciones IVIC, Caracas 2008.

[13] Tutschke, W. and Vanegas, C.J., General algebraic structures of Clifford type andCauchy–Pompeiu formulae for some piecewise constant structure relations, Adv.Appl. Clifford Algebras, 21, 4, 829–838, (2011).

[14] Xu Zhenyuan, A function theory for the operator Dλ, Complex Variables, Theoryand Application: An International Journal, 16, 1, 27–42, (1991)

Eusebio ArizaDepartamento de matematicasUniversidad Simon BolıvarValle de SartenejasCaracas Venezuelae-mail: [email protected]

Antonio Di TeodoroSchool of Mathematics Yachay TechYachay City of Knowledge100119-Urcuquı, Ecuadore-mail: [email protected]





The Fractional Monogenic Signal

Swanhild Bernstein

Abstract. The monogenic signal is a well-known generalization of the analyticsignal. The advantage of such models consists in the fact that they have moreparameters to characterize a signal. In this paper we study two generaliza-tions in R4. Firstly, the fractional Riesz transform and secondly the fractionalmonogenic signal. The Riesz transform is a generalization of the Hilbert trans-form and builds up the monogenic signal of a scalar-valued function f. Thistype of fractional Riesz transform is based on rotations in R4 and not on afractional Fourier transform. Rotations in R4 can by described by quaternionsand therefore the algebraic structure we will use are quaternions.

Mathematics Subject Classification (2010). Primary 30G35; Secondary 44A15.

Keywords. Quaternions, Riesz transform, Hilbert transform, fractional Riesztransform, monogenic signals, fractional Hilbert transform.

1. Introduction

The analytic signal proposed by D. Gabor [7] is a mathematical method to con-struct a unique complex signal associated with a given real signal. The analyticsignal can be characterized by the amplitude, phase and frequency, whereas thereal signal gives just a real number at each point. Driven by applications in op-tics in [9] two generalizations of the Hilbert transform where proposed. In [4] thefractional Hilbert transform was employed for image processing and specifically foredge detection. Later in [3] three different generalizations of Gabor’s analytic signalwere constructed, all of which reduce to the analytic signal when the angle is set tobe π

2 . It was also demonstrated in this work that the fractional Hilbert transformhas the semigroup property, unlike the generalized Hilbert transform in [18].

The monogenic signal proposed in [6] generalizes the analytic signal intohigher dimensions based on Clifford analysis. This approach is different from thehigher-dimensional analytic signal [8]. The monogenic signal has been proven tobe useful in image processing too (see for example [2]). The aim of this paper isto construct a version of a fractional Riesz transform generalizing the fractional

Switzerland

76 S. Bernstein

Hilbert transform and a version of a fractional monogenic signal generalizing thefractional analytic signal in R4 which is embedded into the quaternions. The reasonfor that is that we use a rotation based approach as it is done in [16] for the analyticsignal, fractional Hilbert transform and fractional analytic signal, respectively.Indeed, in the first place, it is not easy to use a fractional Fourier transform inhigher dimensions and, in the second place, quaternions and rotations do fit verywell together. We will also prove some simple properties of our constructions. Wedon’t get a semigroup property because our construction does not depend solelyon angles but also on axes.

The paper is organized as follows. After this introduction in Section 2 weintroduce real and complex quaternions, rotations in R3 and R4, quaternionicanalysis based on some kind of Dirac operator and important properties of mono-genic functions, Hardy spaces and the projections onto these spaces. In Section 3we explain the construction of the analytic signal and the fractional analytic signalas done in [16]. In Section 4 we construct a fractional Riesz operator and finally inSection 5 fractional monogenic signals. Section 6 gives some concluding remarksregarding the 2D case (images), and the situation in R3.

2. Preliminaries

2.1. Quaternions

2.1.1. Real quaternions. Let H denote the skew-field of quaternions with basis{1, i, j, k}. An arbitrary element q ∈ H is given by

q = q01+ q1i+ q2j+ q3k,

where q0, q1, q2, q3 ∈ R. Addition and multiplication by a real scalar definedin component-wise fashion. The multiplication distributes over addition; 1 is themultiplication identity; and

i2 = j2 = k2 = −1,ij = −ji = k, jk = −kj = i, ki = −ik = j.

We decompose a quaternion into two parts that are called its scalar andvector parts. If q = q01+ q1i+ q2j+ q3k, we write

q = q0 + q = Sc (q) + Vec (q),

where q0 = q01 is the scalar part and q = q1i+ q2j+ q3k the vector part of q. OnH a conjugation is defined as q = q01− q for all q ∈ H. The length or norm |q| ofq ∈ H is then

|q| =√q20 + q21 + q22 + q23 =

√qq =

√qq.

All q ∈ H\{0} are invertible with inverse

q−1 =q

|q| .

The Fractional Monogenic Signal 77

The product of two quaternions p, q ∈ H can also be described in terms ofthe usual vector products of vectors in R3. We have

pq = p0q0 + q0p+ p0q + pq

and

pq = −3∑

i=1

piqi + (p1q2 − p2q1)k+ (p2q3 − p3q2)i+ (p3q1 − p1q3)j

= −〈p, q〉+ p× q,

where 〈·, ·〉 denotes the scalar product and × the vector product of vectors in R3.

2.1.2. Complex quaternions. We won’t need much of the theory of complex quater-nions. But because we are using the Fourier transform and will work in Fourierdomain, we have to say something about complex quaternions. A complex quater-nion is given by

q = q01+ q1i+ q2j+ q3k, q0, q1, q2, q3 ∈ C.

The main difference to the real case is that not all elements = 0 are invertible.There are so-called zero-divisors. A simple example are 1+ ik and 1− ik. It turnsout that (1+ ik)(1− ik) = 1− i2k2 = 1− 1 = 0. We will see that zero divisors areimportant for our considerations and won’t create any problems.

2.2. Rotations

We identify H with the four-dimensional Euclidean space R4 by associating q ∈ H

with the vector (q0, q1, q2, q3) ∈ R4.If q ∈ H has |q| = 1, then we call q a unit quaternion. If u is a pure unit

quaternion then u−1 = −u. Two pure unit quaternions u and v are orthogonal ifand only if

〈u, v〉 = −1

2(u v + v u) = 0.

If q is a unit quaternion there is a real number ϕ and a pure unit quaternion usuch that

q = 1 cosϕ+ u sinϕ.

Since u2 = −1, the power series expansion leads to

euϕ =

∞∑n=0

(uϕ)n

n!= 1 cosϕ+ u sinϕ,

providing equivalent representations for a unit quaternion

q = q0 + q = 1 cosϕ+ u sinϕ = euϕ.

It should be mentioned that neither u nor ϕ are uniquely determined. When q =±1 then sinϕ = ±|q| and u = ± q

|q| . When q = ±1, u can be any pure unit

quaternion.

78 S. Bernstein

Let q be a quaternion. Then Lq : H → H and Rq : H → H are defined asfollows

Lq(x) = qx, Rq(x) = xq, x ∈ H.

If q is a unit quaternion, then both Lq and Rq are orthogonal transformations ofH. Thus, for unit quaternions p and q, the mapping Cp,q : H→ H defined by

Cp,q = Lp ◦Rq = Rq ◦ Lp

is also an orthogonal transform of H and Cp1,q1 ◦ Cp2,q2 = Cp1p2,q2q1 .

Theorem 2.1 ([17] and [10]). If q is a unit quaternion, then there exist a pure unitquaternion u and a real scalar ϕ such that q = euϕ. The transform C : R3 → R3

defined by C(x) = qxq is a rotation in the plane orthogonal to u through anangle 2ϕ.

Theorem 2.2 ([17] and [10]). Let p = euϕ and q = evψ , where u and v are pureunit quaternions. The orthogonal transform Cp,q of H is a product of two rotationsin orthogonal planes. If u = ±v, then Cp,q rotates the plane spanned by u+ v anduv−1 through the angle |ϕ+ψ| and the plane spanned by v−u and uv+1 throughthe angle |ϕ − ψ|. If u = ±v, then the invariant planes are the span of 1 and uand its orthogonal complement, and the rotation angles in appropriates planes arestill |ϕ+ ψ| and |ϕ− ψ|.2.3. Quaternionic analysis

The quaternionic analysis presented in this section is based on the nice presentationof Clifford analysis in [5].

2.3.1. Dirac operator. The Dirac operator is defined as the first-order linear dif-ferential operator

Dx = 1∂x0 + i∂x1 + j∂x2 + k∂x3 .

Definition 2.3 (Monogenic functions). Let Ω ⊆ R4 be open and let f be a C1-function in Ω which is (real or complex ) quaternionic-valued. Then f is leftmonogenic or (right monogenic) in Ω if in Ω

Dxf = 0 or fDx = 0.

The connection between monogenic and harmonic functions is due to the factthat Δx = −∂2

x = DxDx.

2.3.2. Integral formulae. Let Ω ⊂ R4 be open, let G be a compact orientablefour-dimensional manifold with boundary ∂G.

Theorem 2.4 (Stokes formula). Suppose that f, g ∈ C1(Ω). Then for each G ⊂ Ω,∫∂G

f(x)n(x)g(x) dS(x) =

∫G

[(fDx)g + f(Dxg)] dx,

where n(x) stands for the outwardly pointing unit normal and dS(x) being thesurface measure.

Because the Dirac operatorDx is a first-order linear differential operator withconstant coefficients there exists a fundamental solution.

Lemma 2.5. A fundamental solution is given by

E(x) =1

2π2

x

|x|4 =1

2π2

(x0 − x1i− x2j− x3k)

|x|4 .

E has the following properties.

1. E is H ∼ R4-valued and belongs to L1loc(R

4).2. E is left and right monogenic in R4\{0}3. and lim

|x|→∞E(x) = 0.

4. DxE = EDx = δ(x), δ(x) being the classical δ-function in R4.

2.3.3. Hardy spaces. Let Σ ∈ R4 be a graph of a Lipschitz function g : R3 → R

and let Ω± be the domains in R4 which lie above, respectively, below Σ, n(y) isthe a.e. on Σ defined outward unit normal at y ∈ Σ and dS(y) the elementarysurface element on Σ.

Definition 2.6 (Hardy spaces). Let 1 0

∫Σ

|F (y ± ε)|p dS(y) <∞}

is the Hardy space of (left) monogenic functions in Ω±.

Definition 2.7 (Integral operators). For f ∈ Lp(Σ) and x ∈ R4\Σ,CΣf(x) =

∫Σ

E(x− y)n(y)f(y) dS(y)

is the Cauchy transform of f.For f ∈ Lp(Σ) and a.e. x ∈ Σ,

RΣf(x) = 2p.v.

∫Σ


= 2 limε→0+

∫y∈Σ:|x−y|>ε


is the Riesz transform (or Hilbert transform) of f .

Remark 2.8. The transform R is in Clifford analysis and function theory usu-ally called Hilbert transform. In signal processing this transform is called Riesztransform to distinguish between the higher-dimensional analytic signal, which isrelated to analytic functions in Cm, and the monogenic signal which is built by theRiesz operators and they form the Clifford–Riesz transform. (The Riesz operatorsare different from the Riesz potentials!)

Theorem 2.9. Let f ∈ Lp(Σ), 1 < p <∞. Then

1. CΣf ∈ Hp(Ω±).2. CΣf has non-tangetial limits (C±Σ ) at almost all x∗ ∈ Σ.

80 S. Bernstein

3. Putting

P+Σf(x

∗) = (CΣf)+(x∗) and P−Σf(x

∗) = −(CΣf)−(x∗)

then P±Σ are bounded projections in Lp(Σ).

4. (Plemelj–Sokhotzki formulae). For a.e. x∗ ∈ Σ,

P+Σf(x

∗) = 12 (f(x

∗) +RΣf(x∗)) and P−

Σf(x∗) = 1

2 (f(x∗)−RΣf(x

∗))

whence1 = P+

Σ + P−Σ and RΣ = P+

Σ − P−Σ .

In particular RΣ is a bounded linear operator on Lp(Σ) and, puttingLp,±(Σ) = P±Lp(Σ), leads to the decomposition into Hardy spaces

Lp(Σ) = Lp,+(Σ)⊕ Lp,−(Σ).

A special situations occurs for Σ = R3. In this case if C denotes the Cauchyintegral and R the Riesz transform, the boundary values of monogenic functionsin Fourier domain can be characterized by the Fourier transform

Ff(ξ) = 1√2π

3

∫R3

ei〈x,ξ〉f(x) dx.

Using that

F(

2xj

2π2|x|4)

= −i ξj|ξ| , j = 1, . . . , 3,

and set [11]

χ±(ξ) = 12

(1± i

ξ

|ξ|),

it should be noticed that

χ2±(ξ) = χ±(ξ) and χ+(ξ) + χ−(ξ) = 1, χ+(ξ)χ−(ξ) = χ−(ξ)χ+(ξ) = 0.

Which means that χ± are projections and zero divisors. The boundary values ofmonogenic function in upper half-space are characterized in the next theorem.

Theorem 2.10 ([11]). For f ∈ Lp(Rm) the following statements are equivalent:

1. The non-tangential limit of CΣf is a.e. equal to f,2. Rf = f,3. Ff = χ+ Ff ,

and characterizes boundary values of monogenic functions.

Specifically for f ∈ Lp(R3) the function f + Rf satisfies R(f + Rf) =Rf +R2f = f +Rf, i.e., f +Rf ∈ Lp,+.

The monogenic signal fM based on the real-valued signal f is defined as

fM := f +Rf

and can be described as a Fourier-integral operator with symbol

2χ+(ξ) =

(1 +

iξ

|ξ|), i.e., fM (x) = F−1

((1 +

iξ

|ξ|)f(ξ)

)


3. The analytic signal

3.1. Hilbert transform

For a real-valued function f(t), its Hilbert transform is defined by

(Hf)(t) = p.v.1

π

∫ ∞

−∞

f(τ)

t− τdτ = (h ∗ f)(t),

where h(t) = 1πt is the Hilbert convolution kernel and p.v. denotes the Cauchy

principal value integral. The Fourier transform of h(t) is h(ξ) = −i sgn(ξ), wheresign(.) denotes the signum function. The analytic signal by Gabor is now given byboundary values of an analytic function in the upper half-space. It is easily seenthat

χ±(ξ) =1

2(1± sgn(ξ))

are projections onto the Hardy spaces. The analytic signal related to f(t) is de-fined as

fA(t) = f(t) + i(Hf)(t)

and in frequency domain

fA(ξ) = (1 + sgn(ξ))f (ξ) = 2χ+(ξ)f(ξ).

The analytic signal operator A can be represented as

A = I + iH.

3.2. Fractional Hilbert operator and analytic fractional signal

The fractional Hilbert kernel hϕ(t) corresponding to the fractional Hilbert operatorHϕ is given in Fourier domain as

hϕ(ξ) = e−iϕχ+(ξ) + eiϕχ−(ξ), −π2 ≤ ϕ ≤ π

2 .

This leads to the fractional Hilbert operator Hϕ defined as

Hϕ = cosϕ I + sinϕH.

For ϕ = 0 we get the identity operator and for ϕ = π2 the fractional Hilbert

operator coincides with the standard Hilbert operator. The analytic fractionaloperator is

fϕA(t) = (Aϕf)(t) = f(t) + ei(π−ϕ)(Hϕf)(t).

4. The fractional Riesz operator

In what follows the function f will be considered to be scalar-valued, i.e., real-

or complex-valued, and therefore the function f and its Fourier transform f willcommute with all quaternions. To construct a fractional Riesz operator we startwith a definition similar to that of the 1D fractional Hilbert operator Hϕ. Wewill work in Fourier domain and replace the projections for analytic functions bythose of monogenic functions and rotate them in R4. The convolution kernel of

82 S. Bernstein

the fractional Riesz transform is given by

rp,q(ξ) = Cp,q−1χ−(ξ) + Cp−1,qχ+(ξ) = euϕ2 χ−(ξ)e−v

ψ2 + e−u

ϕ2 χ+(ξ)e

vψ2

= (cos ϕ2 + u sin ϕ

2 )1

2

(1− iξ

|ξ|)(cos ψ

2 − v sin ψ2 )

+ (cos ϕ2 − u sin ϕ

2 )1

2

(1 +

iξ

|ξ|)(cos ψ

2 + v sin ψ2 )

=1

2

(cos ϕ

2 cos ψ2 − cos ϕ

2 sin ψ2 v + sin ϕ

2 cos ψ2 u− sin ϕ

2 sin ψ2 uv

− cos ϕ2 cos ψ

2

iξ

|ξ| + cos ϕ2 sin ψ

2

iξ

|ξ|v − sin ϕ2 cos ψ

2 uiξ

|ξ| + sin ϕ2 sin ψ

2 uiξ

|ξ|v

+ cos ϕ2 cos ψ

2 + cos ϕ2 sin ψ

2 v − sin ϕ2 cos ψ

2 u− sin ϕ2 sin ψ

2 uv

+cos ϕ2 cos ψ

2

iξ

|ξ| + cos ϕ2 sin ψ

2

iξ


2 uiξ

|ξ| − sin ϕ2 sin ψ

2 uiξ

|ξ|v)

= cos ϕ2 cos ψ

2 − sin ϕ2 sin ψ

2 uv + cos ϕ2 sin ψ

2

iξ


2 uiξ

|ξ|=

1

2

(cos(ϕ2 + ψ

2 ) + cos(ϕ2 − ψ2 )− (cos(ϕ2 − ψ

2 )− cos(ϕ2 + ψ2 ))uv

)+

1

2

((sin(ϕ2 + ψ

2 )− sin(ϕ2 − ψ2 ))

iξ

|ξ|v − (sin(ϕ2 + ψ2 ) + sin(ϕ2 − ψ

2 ))uiξ

|ξ|)

=1

2cos(ϕ2 + ψ

2 )(1 + uv) +1

2cos(ϕ2 − ψ

2 )(1− uv)

+1

2sin(ϕ2 + ψ

2 )

(iξ

|ξ|v − uiξ

|ξ|)− 1

2sin(ϕ2 − ψ

2 )

(iξ

|ξ|v + uiξ

|ξ|)

=1

2cos(ϕ2 + ψ

2 )(1 + uv) +1

2cos(ϕ2 − ψ

2 )(1− uv)

+1

2sin(ϕ2 + ψ

2 )

(−⟨iξ

|ξ| , v − u

⟩+

iξ

|ξ| × (v + u)

)− 1

2sin(ϕ2 − ψ

2 )

(−⟨iξ

|ξ| , v + u

⟩+

iξ

|ξ| × (v − u)

)and hence

rp,q,f =1

2cos(ϕ2 + ψ

2 )(1 + uv)f +1

2cos(ϕ2 − ψ

2 )(1 − uv)f

+1

2sin(ϕ2 + ψ

2 )

(−⟨iξ

|ξ| f , v − u

⟩+

iξ

|ξ| f × (v + u)

)− 1

2sin(ϕ2 − ψ

2 )

(−⟨iξ

|ξ| f , v + u

⟩+

iξ

|ξ| f × (v − u)

).

That is indeed a complicated formula. We look therefore at some special cases.


1. Let be u = v and ϕ = ψ, i.e., p = q and uv = u2 = −1 then

rp,p = 1 + sinϕ

(iξ

|ξ| × u

).

2. Let be v = −u and ϕ = ψ, i.e., p = −q and uv = −u2 = 1 then

rp,−p = cosϕ+ sinϕ

⟨iξ

|ξ| , u⟩

= cosϕ+ sinϕi(u1ξ1 + u2ξ2 + u3ξ3)

|ξ| .

That is the symbol of the operator

Ru,ϕ,−u,ϕf = cosϕf + sinϕ〈u, Rf〉,where 〈u, Rf〉 is the generalized Riesz transform by M. Unser and D. VanDe Ville [15].

4.1. The isoclinic fractional Riesz transform

More promising are the cases of isoclinic rotations. The property of an rotation tobe an isoclinic is nicely explained in [13] and we will cite it. Let P be an arbitrary4D point, represented as a quaternion P = w1+ xi+ yj+ zk. Let p and q be unitquaternions. Consider the left- and right-multiplication mappings P �→ pP andP �→ Pq. Both mappings have the property of rotating all half-lines originatingfrom O through the same angle (arccos p0 and arccos q0 respectively); such rota-tions are denoted as isoclinic. Because the left- and the right-multiplication aredifferent from each other resulting in different rotations we have to distinguish be-tween left- and right-isoclinic rotations. Conversely, an isoclinic 4D rotation aboutO (different from the non-rotation I and from the central reversion −I) is rep-resented by either a left-multiplication or a right-multiplication by a unique unitquaternion. This theorem is presumably due to R.S. Ball; in [1] the author doesnot mention it explicitly as a theorem, but nevertheless gives a proof. However,Ball’s proof is slightly incomplete, a complete proof is given by J.E. Mebius in[12]. Hence, the multiplication with an unit quaternion from the right (or left)only describes a (right- or left-) isoclinic rotation in R4. Moreover, any rotation inR4 is a combination of a left- and a right-isoclinic rotation (see Theorem 2.2).

We consider the case of right isoclinic rotations, i.e., p = e0 = 1 = 1 and q = evψ.In this case we get

rv,ψ = r1,q = C1,q−1χ−(ξ) + C1,qχ+(ξ) = eu·0χ−(ξ)e−vψ + e−u·0χ+(ξ)evψ

=1

2cosψ(1 + uv + 1− uv) +

1

2sinψ

(iξ

|ξ|v − uiξ

|ξ| +iξ

|ξ|v + uiξ

|ξ|)

= cosψ + sinψiξ

|ξ| v

and hence

rv,ψ f = cosψf + sinψiξ

|ξ| f v.

84 S. Bernstein

For ψ = 0 we get rv,0f = f and for ψ = π2 we get rv, π2

f =iξ

|ξ| f v. That is very

similar to the complex case. This leads to the definition of the fractional monogenicsignal.

4.2. Properties of the fractional Riesz operator

Theorem 4.1. Let be f, f1, f2 ∈ Lp(R3), 1 < p <∞, then Ru,ϕf, Ru,ϕf1, Ru,ϕf2 ∈Lp(R3) and Ru,ϕ fulfills the following properties:

(P1) Linearity:

(Ru,ϕ(α1f1 + α2f2))(x) = α1(Ru,ϕf1)(x) + α2(Ru,ϕf2)(x), ∀α1, α2 ∈ C.

(P2) Shift-invariance: Sτ (Ru,ϕf)(x) = Ru,ϕ(Sτf)(x), τ ∈ Rm, where (Sτf)(x) =f(x− τ),

(P3) Scale-invariance: Dσ(Ru,ϕf)(x)=Ru,ϕ(Dσf)(x), σ ∈ R+, where (Dσf)(x) =f(σ−1x),

(P4) Orthogonality:If f, g ∈ L2(R3) such that 〈〈f, g〉〉 = 0, then 〈〈Ru,ϕf, Ru,ϕg〉〉CH = 0. Here,〈〈·, ·〉〉 denotes the usual L2-scalar product and 〈〈·, ·〉〉CH will be defined inRemark 4.2.

Remark 4.2. Let p and q be complex quaternions which can also be interpretedas vectors in C4. As vectors in C4 their scalar product is p · q =

∑3j=0 p

C

j qj andcan be rewritten in terms of the scalar part of a product of complex quaternionsinvolving complex and quaternionic conjugation:

〈p, q〉 = Sc(pCH q

)=

3∑j=0

pCj qj .

Based on that the function space of quaternionic-valued functions, where eachcomponent is an L2-function, can be equipped with the following scalar product

〈〈f, g〉〉CH := Sc

∫R3

fCH

(x) g(x) dx = Sc

⎛⎝ 3∑j=0

∫R3

fC

j (x) gj(x) dx

⎞⎠ ,

which is a complex scalar product for the L2-space of complex quaternionic-valuedfunctions.

Proof: (P1), (P2) and (P3) follow from the linearity, shift-invariance and scale-invariance of the Riesz transform. Property (P4) involves the scalar products of

the corresponding spaces. We use the definition of 〈f , g〉CH and Parseval’s identity

which allows to replace a complex-valued (scalar-valued) function fA(x) by its

Fourier transformˆfA(ξ). We have

〈〈Ru,ϕf, Ru,ϕg〉〉CH = Sc

∫R3

(Ru,ϕf)(x)CH

(Ru,ϕg(x)) dx

= Sc

∫R3

(ru,ϕ(ξ)f(ξ)CH

ru,ϕ(ξ)g(ξ) dξ

= Sc

∫R3

(cosϕ f(ξ)− iξ

|ξ| f(ξ)u sinϕ)CH (

cosϕ g(ξ)− iξ|ξ| g(ξ)u sinϕ

)dξ

= Sc

∫R3

(cosϕ f(ξ)

C

+ uf(ξ)C

iξ|ξ| sinϕ

)(cosϕ g(ξ)− iξ

|ξ| g(ξ)u sinϕ)dξ

= Sc

∫R3

(cosϕ+ u iξ

|ξ| sinϕ)(

cosϕ− iξ|ξ|u sinϕ

)f(ξ)

C

g(ξ) dξ

= Sc

∫R3

(cos2 ϕ− u

(iξ|ξ|)2

u sin2 ϕ

)f(ξ)

C

g(ξ) dξ

=

∫R3

(cos2 ϕ+ sin2 ϕ

)f(ξ)

C

g(ξ) dξ =

∫Rm

f(x)C

g(x) dx = 〈〈f, g〉〉.

It has to be mentioned that this equation is only true for the scalar product takenas the scalar part of the inner product. The adjoint operator

R∗u,ϕ = (cosϕ)f − (sinϕ)u(Rf)

is not the inverse operator but we have that

R∗u,ϕRu,ϕf = (cosϕ I − sinϕuR)(cosϕ I + sinϕRu)f

= cos2 ϕf − sin2 ϕuR2f u− cosϕ sinϕ(u (Rf)− (Rf)u)

= (cos2 ϕ+ sin2 ϕ)f − 12 sin(2ϕ)(u× (Rf))

= f − 12 sin(2ϕ) (u× (Rf))

and thus Sc (R∗u,ϕRu,ϕf) = f.

5. Fractional monogenic signal

Definition 5.1. Let f ∈ Lp(R3), 1 < p < ∞, the fractional monogenic signal withrespect to the right isoclinic rotation q = evψ ∈ H1 is defined in Fourier domain as

fv,ψM (ξ) := f(ξ) + rv,ψ(ξ)f (ξ)e

v(π+ψ).

We demonstrate that the fractional monogenic signal are the boundary values

of a right monogenic function in the upper half-space and hence fv,ψM ∈ Lp,+(R3).

We have

1 + rv,ψ(ξ)ev(π+ψ) = 1 +

(cosψ + sinψ

iξ

|ξ| v)ev(π+ψ)

= 1−(cosψ + sinψ

iξ

|ξ| v)(cosψ + v sinψ)

= 1− cos2 ψ − v cosψ sinψ − sinψ cosψiξ

|ξ|v + sin2 ψiξ

|ξ|

= sin2 ψ

(1 +

iξ

|ξ|)− sinψ cosψ

(1 +

iξ

|ξ|)v

86 S. Bernstein

= sinψ

(1 +

iξ

|ξ|)(sinψ − v cosψ)

= sinψ

(1 +

iξ

|ξ|)(− cos

(π2 + ψ

)− v(sin(π2 + ψ

)))= χ+(ξ)

(− sinψ ev(

π2 +ψ)

).

5.1. Properties of the fractional monogenic signal

The fractional monogenic signal is given by

fu,ϕM (x) = (Mu,ϕf)(x), where Mu,ϕ = I +M eu(π−ϕ)Ru,ϕ.

The fractional monogenic signal are the boundary values of an right-monogenicfunction in the upper half-space and the amplitude of the fractional monogenicsignal is the amplitude of the monogenic signal modulated by | sinψ|, i.e.,

|fu,ϕM (x)| = | sinϕ||fM|.

The phase of the fractional monogenic signal is also different from that of themonogenic signal because the scalar part of the fractional monogenic signal is acombination of the function f and the scalar product of the Riesz transform of f(considered as a vector) and the vector u.

Theorem 5.2. Let be f, f1, f2 ∈ Lp(R3), 1 < p <∞, then Ru,ϕf, Ru,ϕf1, Ru,ϕf2 ∈Lp(R3) and Ru,ϕ fulfills the following properties:

(M1) Linearity:

(Mu,ϕ(α1f1 +α2f2)(x) = α1(Mu,ϕf1)(x) +α2(Mu,ϕf2))(x), ∀α1, α2 ∈ C.

(M2) Shift-invariance: Sτ (Mu,ϕf)(x) =Mu,ϕ(Sτf)(x), τ ∈ Rm,

(M3) Scale-invariance: Dσ(Mu,ϕf)(x) =Mu,ϕ(Dσf)(x), σ ∈ R+,

6. Concluding remarks

It is also of some interest to consider a monogenic signal of two variables (forimages). In this case

fM(x, y) = f(x, y) + i(R1f)(x, y) + j(R2f)(x, y).

That can be interpreted as something living in R3 and it should be even easier toconsider rotations for this case. A rotation in R3 can be described by quaternionsin the following way [10]. Any rotation in R3 can be described by the mapping

R3 → R3, r �→ ara−1,

where a is a unit quaternion and a = cos( |a|2 )+ a|a| sin(

|a|2 ) = euϕ with u = a

|a| and

ϕ = |a|2 . The first problem arises from the fact that (I +R)f is not a vector in R3

even though it can be identified with a vector. The second problem consists in thefact that the rotations in R3 are defined by multiplication from the right and theleft which will destroy monotonicity. Therefore we suggest to embed the problem


into R4 and use a Riesz operator R = iR1 + jR2 and a unit vector u = iu1 + ju2.Then the definition

ru,ϕ(ξ) = χ−(ξ)e−vψ2 + χ+(ξ)e

vψ2 = cos ψ

2 + sin ψ2

iξ

|ξ| v

= cos ψ2 + sin ψ

2

⟨iξ

|ξ| , v⟩+ sin ψ

2

(iξ

|ξ| × v

)seems to give a useful definition for a fractional Riesz operator in this case. Alsohigher dimensions are of interest. Here, the description of a rotation is even morecomplicated. A way around could be to look for isoclinic and pseudo-isoclinicrotations and their behavior [14]. Another possibility to define a fractional Riesztransform for 2D signals (images) is to embed the signal f as fk into H. Then the

projections 12

(1± iξ

|ξ|k)multiplied with fk can be identified with vectors in R3

and the vector can be rotated in R3 using quaternions.

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[10] P. Lounesto, Clifford Algebras and Spinors, Cambridge Univ. Press, 1997.

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[12] J.E. Mebius, Applications of quaternions to dynamical simulation, computer graphicsand biomechanics, Ph.D. Thesis Delft University of Technology, Delft, 1994.

88 S. Bernstein

[13] J.E. Mebius, A Matrix-based Proof of the Quaternion Representation Theorem forFour-Dimensional Rotations, http://arxiv.org/abs/math/0501249v1.

[14] A. Richard, L. Fuchs, E. Andres, G. Largeteau-Skapin, Decomposition of nD-rotations: classification, properties and algorithm, Graphical Models, 73(6) (2011),346–353.

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Swanhild BernsteinTU Bergakademie FreibergInstitute of Applied AnalysisD-09599 Freiberg, Germanye-mail: [email protected]

http://arxiv.org/abs/math/0501249v1


Weighted Bergman Spaces

Luıs Javier Carmona L., Lino Feliciano Resendis Ocampoand Luis Manuel Tovar Sanchez

Abstract. In this paper we study weighted Bergman spaces, through Greenfunction and Mobius transformations, and its relationship and remarkabledifferences with the F (p, q, s) Zhao spaces and so with other classical weightedfunction spaces.

Mathematics Subject Classification (2010). Primary 30C45; Secondary 30J.

Keywords. Bergman spaces, F (p, q, s) Zhao spaces.

1. Introduction

Let 0 < r <∞. Define Dr(a) := {z ∈ C : |z−a| < r}, Dr = Dr(0) and Ar = D\Dr.We denote by D = D1 the open unit disk in the complex plane C.

Let ϕz : C \ { 1z} → C be the Mobius transformation,

ϕz(w) =z − w

1− zw,

with pole at w = 1/z that verifies ϕ−1z = ϕz and

1− |ϕz(w)|2 =(1− |z|2)(1− |w|2)

|1− zw|2 = (1 − |w|2)|ϕ′z(w)| . (1.1)

For z, w ∈ D, we denote the Green function of D, with logarithmic singularity atz, by

g(w, z) = ln|1− zw||z − w| = ln

1

|ϕz(w)| . (1.2)

Let H be the space of analytic functions f : D→ C. For 0 < p <∞, −2 < q <∞,0 ≤ s <∞, let f ∈ H be such that,

hp,q,s(f)(z) :=

∫D

|f(w)|pgs(w, z) dAq(w) <∞, (1.3)

where dAq(w) := (1− |w|2)qdA(w).This work was completed with the support of Conacyt.

Switzerland

90 L.J. Carmona L., L.F. Resendis O. and L.M. Tovar S.

We define the q, s-weighted p-Bergman space as

sApq := { f ∈ H : sup

z∈D

hp,q,s(f)(z) <∞ }

and for 0 < s <∞, the little q, s-weighted p-Bergman space as

s,0Apq := { f ∈ H : lim

|z|→1−hp,q,s(f)(z) = 0 } .

In the same way, for 0 < p <∞, −2 < q <∞ and 0 ≤ s <∞, let f ∈ H be suchthat,

lp,q,s(f)(z) :=

∫D

|f(w)|p(1− |ϕz(w)|2)s dAq(w) <∞ (1.4)

and define

L(p, q, s) := { f ∈ H : supz∈D

lp,q,s(f)(z) <∞ }

and for 0 < s <∞L0(p, q, s) := { f ∈ H : lim

|z|→1−lp,q,s(f)(z) = 0 } .

We write Lp = L(p, 0, 0) and observe that 0A2q = 0L

2q = A2

q is the classicalBergman space of analytic functions. Following Zhu ([8]), we refer also the spacesAp

q as the classical Bergman spaces. We define further

‖ f ‖g =‖ f ‖g,p,q,s= supz∈D

(hp,q,s(f)(z))1p

and

‖ f ‖ϕ =‖ f ‖ϕ,p,q,s= supz∈D

(lp,q,s(f)(z))1p .

Let 0 < α <∞. We say that f ∈ H belongs to the α-Bloch–Bergman space Bα if

‖ f ‖α= supz∈D

(1 − |z|2)α|f(z)| <∞

and belongs to the little α-Bloch–Bergman space Bα0 if

‖ f ‖α= lim|z|→1−

(1 − |z|2)α|f(z)| = 0 .

It is clear that Bα0 ⊂ Bα.

The aim of this paper is to obtain explicitly properties of the weightedBergman spaces, in particular we study the nested scale between the classicalBergman spaces and the Bloch–Bergman spaces. We also study the relation-ships with the F (p, q, s) spaces introduced by Ruhan Zhao, (see [6]). QuaternionicBergman spaces will be studied in a forthcoming paper [4].

Weighted Bergman Spaces 91

2. The α-Bloch–Bergman space

The sets Bα are vectorial spaces and is immediate consequence of an argumentof normality that Bα and Bα

0 are complete spaces with the norm ‖ ‖α; moreoverconvergence in this norm implies uniform convergence by compact sets on the unitdisk.

Let 0 < R < 1. The pseudohyperbolic disk is defined by

D(z,R) := ϕz(DR) = { w ∈ D : |ϕz(w)| < R } .

In fact D(z,R) is a Euclidean disk with center and radius given by

c =1−R2

1−R2|z|2 z, R =1− |z|2

1−R2|z|2R (2.1)

and we denote by |D(z,R)| its area.We need the following results:

Lemma 2.1 ([8]). Let t > −1, c ∈ R. Define It,c : D→ R by

It,c(z) =

∫D

(1− |w|2)t|1 − zw|2+t+c

dA(w)

and Jc : D→ R by

Jc(z) =

∫ 2π

0

dθ

|1− ze−iθ|1+c

Then

(i) If c < 0, then It,c(z) ≈ Jc(z) ≈ 1 when |z| → 1−.

(ii) If c = 0, then It,c(z) ≈ Jc(z) ≈ ln1

1− |z|2 , when |z| → 1−.

(iii) If c > 0, then It,c(z) ≈ Jc(z) ≈ 1

(1− |z|2)c .

As a consequence of the previous lemma we have

Lemma 2.2. Let −2 < q <∞ and 0 < s <∞. Then

lim|z|→1−

∫D

gs(w, z) dAq(w) = 0

lim|z|→1−

∫D

(1− |ϕz(w)|2)s dAq(w) = 0 .

Proof. By the change of variable w = ϕz(v) we have∫D

gs(w, z) dAq(w) =

∫D

lns1

|v| (1 − |ϕz(v)|2)q|ϕ′z(v)|2 dA(v)

= (1 − |z|2)q+2

∫D

lns 1

|v|(1− |v|2)q+2

|1− zv|4+2qdA(v) .

The result follows by taking t = q + 2 in the previous lemma. The other proof issimilar. �

The α-Bloch–Bergman space has the following characterizations:

Theorem 2.3. Let 0 < p < ∞, 0 ≤ α < ∞, 0 < r < 1 < s < ∞ and f ∈ H. Thefollowing quantities are equivalent, that is, if one of them is finite then the otherones are finite too:

(i) supz∈D

(1− |z|2)pα|f(z)|p;

(ii) supz∈D

1

|D(z, r)|1− pα2

∫D(z,r)

|f(w)|p dA(w);

(iii) supz∈D

∫D(z,r)

|f(w)|p dApα−2(w);

(iv) supz∈D

∫D

|f(w)|p(1− |ϕz(w)|2)s dApα−2(w);

(v) supz∈D

∫D

|f(w)|pgs(w, z) dApα−2(w);

(vi) supz∈D

∫D

|f(w)|p(log

1

|w|)pα

|ϕ′z(w)|2 dA(w).

Proof. The subharmonicity of |f(z)|p for 0 < p < ∞ is the key of the proof. Theproof in general is similar to the proof Theorem 1 of R. Zhao in [6]. �

An analogous result is true with the little α-Bloch–Bergman space, (see The-orem 2 of R. Zhao in [6]).

Since we will need some estimations, we include a proof of the equivalence of(i) and (v). Analogous proof can be given for the equivalence of (i) and (iv).

Theorem 2.4. Let 0 < p <∞, −2 < q <∞. If 0 < s <∞, then sApq ⊂ B

q+2p and

s,0Apq ⊂ B

q+2p

0 . If 1 < s <∞, then sApq = B

q+2p and s,0Ap

q = Bq+2p

0 .

Proof. Let f ∈ sApq and 0 < R < 1 be fixed. Then by the change of variable

w = ϕz(ζ) and subharmonicity we have∫D

|f(w)|pgs(w, z) dAq(w) ≥∫D(z,R)

|f(w)|pgs(w, z) dAq(w)

≥ lns1

R

∫DR

|f(ϕz(ζ))|p(1− |ϕz(ζ)|2)q|ϕ′z(ζ)|2 dA(ζ)

≥ 1

22q+4lns

1

R(1− |z|2)q+2

∫ R

0

(1 − ρ2)qρ

∫ 2π

0

|f(ϕz(ρeiθ))|p dθdρ

≥ 1

22q+4lns

1

R(1− |z|2)q+2|f(z)|p

∫ R

0

(1 − ρ2)qρ dρ

= C(R)(1 − |z|2)q+2|f(z)|p (2.2)

and the first two claims follow from the previous estimation.

Now, if f ∈ Bq+2p , again by using the change of variable w = ϕz(ζ)∫

D

|f(w)|pgs(w, z) dAq(w) ≤‖ f ‖pq+2p

∫D

1

(1− |w|2)2 gs(w, z) dA(w)

=‖ f ‖pq+2p

∫D

1

(1− |ϕz(ζ)|2)2 lns1

|ζ| |ϕ′z(ζ)|2 dA(ζ)

=‖ f ‖pq+2p

∫D

1

(1− |ζ|2)2 lns1

|ζ| dA(ζ) ;

so as 1 < s <∞, the last integral is finite (see [6], Lemma 3.2).

Let f ∈ Bq+2p

0 . Given ε > 0, there exists 0 < R < 1 such that

|f(w)|p ≤ ε

(1− |w|2)q+2for all w ∈ AR.

Thus ∫AR

|f(w)|pgs(w, z) dAq(w) ≤ ε

∫AR

gs(w, z)

(1 − |w|2)2 dA(w)

and ∫DR

|f(w)|pgs(w, z) dAq(w) ≤ maxw∈DR

|f(w)|p∫DR

gs(w, z) dAq(w)

and we conclude the proof applying Lemma 2.2. �If we write the estimation (2.2), with s = 0, we obtain that the classical

Bergman spaces are included in some α-Bloch–Bergman space.

Corollary 2.5. Let 0 < p <∞, −2 < q <∞. Then Apq ⊂ B

q+2p

0 and ‖ ‖ q+2p≤ C ‖ ‖g.

Proof. By (2.1), |D(z,R)| → 0 when |z| → 1− then apply absolute continuity ofthe integral. �

3. Properties of sApq and L(p, q, s)

In this section we obtain basic properties of the Bergman spaces sApq and L(p, q, s).

Let 0 < p <∞ and 0 < s < s′ <∞. It is immediate that

L(p, q, s) ⊂ L(p, q, s′) and L0(p, q, s) ⊂ L0(p, q, s′) ;

since 1− x2 ≤ 2 ln1

xfor all x ∈ (0, 1] then

sApq ⊂ L(p, q, s) and s,0Ap

q ⊂ L0(p, q, s) .

It is easy verify that s,0Apq is a vectorial space.

For 0 < p < 1, ‖ ‖pg and ‖ ‖pϕ define metrics and for 1 ≤ p < ∞, ‖ ‖g and‖ ‖ϕ define norms.

Theorem 3.1. Let 0 < p < ∞, −2 < q < ∞ and 0 < s < ∞, with −1 < q + s.If f, g ∈ L(p, q, s) (or L0(p, q, s)), then λf + ηg ∈ L(p, q, s) (or L0(p, q, s)) forλ, η ∈ C. Moreover, if 0 < p < r <∞, then L(r, q, s) ⊂ L0(p, q, s).

Proof. Define for each measurable set E ⊂ D,

νz,q,s(E) =

∫E

(1 − |ϕz(w)|2)s dAq(w) .

Thus νz,q,s(D) <∞ by Lemma 2.1 and by Lemma 2.2 we have

lim|z|→1−

νz,q,s(D) = 0 .

Moreover νz,q,s is absolutely continuous with respect to the Lebesgue measure onD and ∫

D

|f(w)|p dνz,q,s(w) =∫D

|f(w)|p(1− |ϕz(w)|2)s dAq(w) .

Let f, g ∈ L(p, q, s). The theorem will follow from the next estimation

|λf(w) + ηg(w)|p ≤ 2p(|λ|p|f(w)|p + |η|p|g(w)|p)and the Holder inequality (see [2], Theorem 13.17)(∫

D

|f(w)|p dνz,q,s(w)) 1

p

≤(∫

D

|f(w)|r dνz,q,s(w)) 1

r

νz,q,s(D)1p− 1

r . �

Theorem 3.2. Let 1 ≤ p < ∞, −2 < q < ∞ and 0 < s < ∞. Then ‖ ‖g and‖ ‖ϕ define norms on sAp

q and L(p, q, s), respectively. With these norms sApq and

L(p, q, s) become Banach spaces.

Proof. It is immediate that ‖ · ‖g and ‖ · ‖ϕ are norms.

We prove only that sApq is complete. Let {fn} be a Cauchy sequence in sAp

q .

By Corollary 2.5, {fn} is also a Cauchy sequence in Bq+2p . Then {fn} converges

uniformly on compact sets to f ∈ Bq+2p .

Given ε > 0, there exists N > 0 such that if n ≥ m ≥ N we have∫D

|fn(w)− fm(w)|pgs(w, z) dAq(w) ≤‖ fn − fm ‖pg<( ε2

)p.

By Fatou’s Lemma∫D

|f(w) − fm(w)|pgs(w, z) dAq(w)

≤ lim infn→∞

∫D

|fn(w) − fm(w)|pgs(w, z) dAq(w) ≤( ε2

)p.

Taking the supremum with respect to z ∈ D we see that f − fm ∈ sApq , therefore

f = (f − fm) + fm ∈ sApq . Moreover ‖ f − fm ‖g≤ ε

2 if m ≥ N . �

The following result tells us that L(p, q, s) or sApq can be trivial.

Proposition 3.3. Let 0 < p < ∞, −2 < q < ∞ and 1 < s < ∞ with q + s ≤ −1.Then the space L(p, q, s) = sAp

q and consists only of the constant function 0.

Proof. Let f ∈ L(p, q, s) be a nonzero function. Let 0 < b < 1 be fixed. Since|f(z)|p is a subharmonic function,

lp,q,s(f)(0) =

∫D

|f(w)|p(1 − |w|2)s dAq(w)

≥∫ 1

b

∫ 2π

0

|f(reiθ)|p(1 − r2)q+sr dθ dr

≥∫ 2π

0

|f(beiθ)|p dθ∫ 1

b

(1− r2)q+sr dr =∞ .

So we get a contradiction. The other proof is similar. �

Define

H∞ = { f ∈ H : f is bounded } .

The spaces s,0Apq are not empty, since by Lemma 2.2

H∞ ⊂⋂{ s,0Ap

q : 0 < p <∞, −2 < q <∞, 0 < s <∞, −1 < q + s} .

Thus we can improve Corollary 2.5.

Theorem 3.4. Let 0 0. By Corollary 2.5, there exists 0 < R < 1 such that

(1− |w|2)q+2|f(w)|p < ε for all w ∈ AR (3.1)

and by absolute continuity of the integral∫AR

|f(w)|p dAq(w) < ε . (3.2)

We split the integral

hp,q,s(f)(z) =

∫DR

|f(w)|pgs(w, z) dAq(w) +

∫AR

|f(w)|pgs(w, z) dAq(w) .

By Lemma 2.2 the first integral goes to 0 when |z| → 1−. We split again the secondintegral: By (2.1) we can choose R′, such that 0 < R < R′ < 1 and D(z,R′) ⊂ AR.Now by (3.2) we have∫

AR\D(z,R′)|f(w)|pgs(w, z) dAq(w) ≤ lns 1

R

∫AR\D(z,R′)

|f(w)|p dAq(w)

< ε lns1

R;

by (3.1) and the change of variable w = ϕz(ζ) we have∫D(z,R′)

|f(w)|pgs(w, z) dAq(w) ≤∫D(z,R′)

ε

(1− |w|2)q+2gs(w, z) dAq(w)

= ε

∫DR′

1

(1− |ϕz(ζ)|2)2 lns 1

|ζ| |ϕ′(ζ)|2 dA(ζ)

≤ ε

∫D

1

(1 − |ζ|2)2 lns1

|ζ| dA(ζ)

and the last integral is finite. This concludes the proof. �

We will need the following elementary estimations.

Lemma 3.5. Let q ∈ R and z ∈ D. Then for all w ∈ D

1

ρ(z, q)(1− |w|2)q ≤ (1− |ϕz(w)|2)q ≤ ρ(z, q)(1− |w|2)q, (3.3)

and1

ρ(z, q)≤ |ϕ′

z(w)|q ≤ ρ(z, q) (3.4)

where

ρ(z, q) =

(1 + |z|1− |z|

)|q|.

The spaces sApq and L(p, q, s) are Mobius invariant in the following sense:

Proposition 3.6. Let z ∈ D and 0 < p < ∞, −2 < q < ∞ and 0 ≤ s < ∞. Iff ∈ sAp

q (or f ∈ L(p, q, s)) then f ◦ ϕz ∈ sApq (or f ◦ ϕz ∈ L(p, q, s)), that is,

supb∈D

∫D

|f(ϕz(w))|p gs(w, b) dAq(w) <∞ .

Proof. Let z ∈ D be fixed. We denote by w = ϕz(v), b = ϕz(c). Since the Greenfunction is conformally invariant, g(ϕz(v), ϕz(c)) = g(v, c). Then by (3.3) and (3.4)∫

D

|f(ϕz(w))|p gs(w, b) dAq(w)

=

∫D

|f(ϕz(ϕz(v)))|p|ϕ′z(v)|2(1 − |ϕz(v)|2)qgs(ϕz(v), ϕz(c)) dA(v)

≤ ρ(z, |q|+ 2)

∫D

|f(v)|p(1− |v|2)qgs(v, c) dA(v)

= ρ(z, |q|+ 2)

∫D

|f(v)|pgs(v, c) dAq(v).

Taking the supremum the result follows. �

Observe that, the previous result is also true for the little spaces.We will need the following result.

Lemma 3.7. Let 0 < p <∞, −2 < q <∞, 0 ≤ s <∞ and z ∈ D. Then

1

ρ(z, |q|+ 2)lp,q,s(f)(z) ≤ lp,q,s(f ◦ ϕz)(0) ≤ ρ(z, |q|+ 2)lp,q,s(f)(z) .

Proof. Suppose that lp,q,s(f)(z) <∞. Then by the change of variable formula withv = ϕz(w), we have∫

D

|f(ϕz(v))|p(1− |v|2)q+s dA(v)

=

∫D

|f(w)|p|ϕ′z(w)|2(1− |ϕz(w)|2)q+s dA(w)

≤ ρ(z, |q|+ 2)

∫D

|f(w)|p(1− |w|2)q(1− |ϕz(w)|2)s dA(w).

Conversely, if lp,q,s(f ◦ ϕz)(0) <∞, then with v = ϕz(w):∫D

|f(ϕz(v))|p(1− |v|2)q+s dA(v)

=

∫D

|f(w)|p|ϕ′z(w)|2(1 − |ϕz(w)|2)q+s dA(w)

≥ 1

ρ(z, |q|+ 2)

∫D

|f(w)|p(1− |w|2)q(1− |ϕz(w)|2)s dA(w). �

The following results clarify the relationship between L0(p, q, s) and L(p, q, s).

Lemma 3.8. Let 0 0, there exists 0 < R′ < 1 such that for each z ∈ DR∫AR′|f(w)|p lns 1

|ϕz(w)| dAq(w) < ε .

Proof. As f ∈ Apq+s, given ε > 0, there exists 0 < R′′ < 1 such that∫

AR′′|f(w)|p dAq+s(w) < ε

(1−R)2s

2s.

Likewise there exists α ∈ (0, 1) such that

1− t < ln1

t< 2(1− t), α < t < 1,

thus

(1− r2) < ln1

r2< 2(1− r2), α < r2 < 1.

There exists 1 > R′ > R′′ > 0, such that2(1− |w|2)(1−R)2

< 1 − α for all w ∈ AR′ .

Thus α < |ϕz(w)|2 < 1, for all z ∈ DR and w ∈ AR′ . Then∫AR′

|f(w)|p lns 1

|ϕz(w)| dAq(w)

≤ 2s

(1−R)2s

∫AR′′

|f(w)|p(1− |w|2)s dAq(w) < ε . �

Proposition 3.9. Let 0 < p < ∞, −2 < q < ∞ and 0 ≤ s < ∞. Suppose thatf ∈ H is such that hp,q,s(f)(z) <∞ for all z ∈ D. Then hp,q,s(f) is a continuousfunction on D.

Proof. Let R and R′ be as in Lemma 3.8. Let {bn} ⊂ DR be a sequence such thatbn → z ∈ DR when n→∞. By (2.1), we observe that⋃

b∈DR

ϕb(DR′) = DR′′ , where R′′ =R+R′

1 +RR′ .

Let In : DR′′ → R be defined by

In(w) = |f(ϕbn(w))|p lns1

|w| |ϕ′bn(w)|2(1− |ϕbn(w)|2)q χϕbn(DR′ )(w) ,

where χ denotes the characteristic function. Thus In(w) → Iz(w) if n → ∞.Moreover |In(w)| ≤ M lns 1

|w| for all w ∈ DR′′ and some constant M > 0. Taking

the change of variable w = ϕbn(v) we have∫DR′

|f(w)|p lns1

|ϕbn(w)|dAq(w) =

∫ϕbn (DR′ )

|f(ϕbn(v))|p lns1

|v|· |ϕ′

bn(v)|2(1− |ϕbn(v)|2)q dA(v) .Then by Lebesgue’s theorem

limn→∞

∫DR′′

In(w) dA(w) =

∫DR′′

Iz(w) dA(w) .

After the previous lemma, this concludes the proof. �

Proposition 3.10. Let 0 < p < ∞, −2 < q < ∞, 0 ≤ s < ∞. Suppose that f ∈ His such that lp,q,s(f)(z) <∞ for all z ∈ D. Then lp,q,s(f) is a continuous functionon D.

Proof. If f = 0 on D, it is clear that lp,q,s(f) is continuous. Therefore, we supposethat f = 0, in particular lp,q,s(f)(0) = 0. Let z ∈ D be fixed and let δ > 0 be such

that Dδ(z) ⊂ D. The function l : D× Dδ(z)→ R defined by

(w, ζ)→ (1 − |ζ|2)s|1− ζw|2s

is uniformly continuous on D × Dδ(z). Then given ε > 0, there exists ρ > 0 suchthat if |w′ − w| < ρ and |ζ′ − ζ| < ρ then

|l(w′, ζ′)− l(w, ζ)| < ε

lp,q,s(f)(0),

therefore if |z − z′| < ρ then

|lp,q,s(f)(z)− lp,q,s(f)(z′)|

≤∫D

|f(w)|p (1− |w|2)s |l(w, z)− l(w, z′)| dAq(w) < ε . �

Corollary 3.11. Let 0 < p < ∞, −2 < q < ∞, 0 ≤ s < ∞. Then L0(p, q, s) ⊂L(p, q, s) and s,0Ap

q ⊂ sApq.

Proof. If f = 0 it is clear. Suppose f = 0 and f ∈ L0(p, q, s). Then there exists0 < R < 1 such that lp,q,s(f)(z) < lp,q,s(f)(0) for all R < |z| < 1. By Proposition3.10, lp,q,s(f) attains its finite maximum on DR, then f ∈ L(p, q, s). The otherproof is similar. �Corollary 3.12. Let 0 < p < ∞, −2 < q < ∞ and 0 ≤ s < ∞. If f ∈ L0(p, q, s),then lp,q,s(f) is a uniformly continuous function on D. If f ∈ s,0Ap

q , then hp,q,s(f)

is a uniformly continuous function on D.

Considering the estimations in the proof of Theorem 3.2, we get the followingresult.

Corollary 3.13. Let 1 ≤ p < ∞, −2 < q < ∞ and 0 < s < ∞ with −1 <q+s. Then, s,0Ap

q and L0(p, q, s) are linear Banach subspaces of sApq and L(p, q, s)

respectively.

Proof. It is enough to prove that s,0Apq is a closed subspace of sAp

q . Let {fn} ⊂s,0Ap

q be a sequence converging to f . By Theorem 3.2, f ∈ sApq . Therefore for any

ε > 0, there exists N ∈ N, such that ‖ f − fn ‖g< ε

2for all n ≥ N . By hypothesis

there exists a 0 < R < 1 such that if z ∈ AR then∫D

|fN(w)|pgs(w, z) dAq(w) <εp

2p.

Thus(∫D

|f(w)|pgs(w, z) dAq(w)

) 1p

≤(∫

D

|f(w) − fN(w)|pgs(w, z) dAq(w)

) 1p

+

(∫D

|fN(w)|pgs(w, z) dAq(w)

) 1p

<ε

2+

ε

2= ε .

The other proof is similar. �Remark 3.14. For 0 < p < 1 we get similar results with the respective metrics.

4. The equality sApq = L(p, q, s)

In this section we prove the equality between the spaces sApq and L(p, q, s).

Theorem 4.1. Let 0 < p <∞, −2 < q <∞ and f ∈ H. Suppose that lp,q,s(f)(0) <∞, then:

a) For 0 < s <∞, we have∫D

|f(w)|p lns 1

|w| dAq(w) ≤ t

∫D

|f(w)|p(1 − |w|2)s dAq(w), (4.1)

where t = t(q, s, R) for some 0 < R < 1 fixed.b) For 0 < s < 1, we have∫

D

|f(w)|p|w|−2s(1− |w|2)s dAq(w) ≤ t

∫D

|f(w)|p(1− |w|2)s dAq(w) (4.2)

where t = t(q, s, R) for some 0 < R < 1 fixed.

Proof. a) We follow the idea of the proof given in Theorem 2.2 of Aulaskari et al.[1] with all its details. Let c = . 0183403 be the root of − lnx = 4(1 − x2). Letc < R < 1 be fixed. Define

0 <1

τ(q, s, R)=

∫ R

c

(1− r2)q+sr dr

=1

2(1 + q + s)((1 − c2)1+q+s − (1−R2)1+q+s)

=1

2(1 + q + s)(.9996641+q+s − (1−R2)1+q+s) .

As |f |p is subharmonic

1

τ(q, s, R)

∫ 2π

0

∣∣f(ceiθ)∣∣p dθ =

∫ R

c

(1− r2)q+sr dr

∫ 2π

0

∣∣f(ceiθ)∣∣p dθ

≤∫ R

c

(1− r2)q+sr dr

∫ 2π

0

∣∣f(reiθ)∣∣p dθ

=

∫DR\Dc

|f(w)|p(1− |w|2)s dAq(w) .

Therefore ∫ 2π

0

|f(ceiθ)|p dθ ≤ τ(q, s, R)

∫DR

|f(w)|p(1− |w|2)s dAq(w) . (4.3)

Define

0 < τ(q, s) =

∫ c

0

r(1 − r2)q lns1

rdr. (4.4)

By subharmonicity and (4.3), we have the estimation∫Dc

|f(w)|p(1− |w|2)q lns 1

|w| dA(w)

=

∫ c

0

∫ 2π

0

|f(reiθ)|pr(1 − r2)q lns1

rdθ dr

≤∫ c

0

∫ 2π

0

|f(ceiθ)|pr(1 − r2)q lns1

rdθ dr

=

∫ c

0

r(1 − r2)q lns 1

rdr

∫ 2π

0

|f(ceiθ)|p dθ

≤ τ(q, s, R)τ (q, s)

∫DR

|f(w)|p(1− |w|2)s dAq(w)

≤ τ(q, s, R)τ (q, s)

∫D

|f(w)|p(1− |w|2)s dAq(w).

From the inequality

− lnx ≤ 4(1− x2) for each x ∈ (c, 1], (4.5)

we have ∫D\Dc

|f(w)|p(1 − |w|2)q lns 1

|w| dA(w)

≤ 4s∫D\Dc

|f(w)|p(1− |w|2)s dAq(w)

≤ 4s∫D

|f(w)|p(1− |w|2)s dAq(w).

(4.6)

Let t(q, s, R) = τ(q, s, R)τ (q, s) + 4s. Combining (4.5) and (4.6), we have∫D

|f(w)|p(1− |w|2)q lns 1

|w| dA(w) ≤ t(q, s, R)

∫D

|f(w)|p(1 − |w|2)s dAq(w).

b) For 0 < s < 1, we need to consider instead of (4.4), the following equality

0 <

∫ c

0

r1−2s(1− r2)q+s dr =1

2B[c2, 1− s, 1 + q + s],

where B denotes the incomplete Beta function. Then we prove the formula (4.2)in a similar way to (4.1). �

Theorem 4.2. Let 0 < p <∞, −2 < q <∞ and 0 ≤ s <∞. Then sApq = L(p, q, s).

Proof. We have

1− x2 ≤ −2 lnx for each x ∈ (0, 1] .

Taking x = ϕz(w) we have 1− |ϕz(w)|2 ≤ 2g(w, z). Therefore

lp,q,s(f)(z) ≤ 2hp,q,s(f)(z) for each z ∈ D . (4.7)

Thus sApq ⊂ L(p, q, s).

We prove now that L(p, q, s) ⊂ sApq . For this, let f ∈ L(p, q, s), then lp,q,s(f)(z) <

∞. By hypothesis and Lemma 3.7, lp,q,s(f ◦ ϕ)(0) <∞. Since |f ◦ ϕa|p is subhar-monic, the formula (4.1) reads∫

D

|f(ϕz(w)) lns 1

|w| dAq(w) ≤ t(q, s, R)

∫D

|f(ϕz(w))|p(1− |w|2)s dAq(w) .

Consider the change of variable w = ϕz(v) to obtain∫D

|f(v)|p|ϕ′z(v)|2(1− |ϕz(v)|2)q lns 1

|ϕz(v)| dA(v)

≤ t(q, s, R)

∫D

|f(v)|p|ϕ′z(v)|2(1 − |ϕz(v)|2)q+s dA(v) ,

or, equivalently,

0 ≤∫D

|f(v)|p|ϕ′z(v)|2(1 − |ϕz(v)|2)q

·(t(q, s, R)(1− |ϕz(v)|2)s − lns

1

|ϕz(v)|)

dA(v)

≤ ρ(z, |q|+ 2)

∫D

|f(v)|p(1 − |v|2)q

·(t(q, s, R)(1− |ϕz(v)|2)s − lns

1

|ϕz(v)|)

dA(v)

then we obtain∫D

|f(v)|p lns 1

|ϕz(v)| dAq(v) ≤ t(q, s, R)

∫D

|f(v)|p(1− |ϕz(v)|2)s dAq(v), (4.8)

and the theorem follows. �

Corollary 4.3. Let 0 < p < ∞, −2 < q < ∞ and 0 ≤ s < ∞. Then s,0Apq =

L0(p, q, s).

Proof. It is a consequence of the formulas (4.7) and (4.8) since

lp,q,s(f)(z) ≤ 2hp,q,s(f)(z) ≤ 2t(q, s, R)lp,q,s(f)(z) . �

Observe that we have used a completely different idea in the proof of theprevious theorem to the used by Zhao in Theorem 2.4 of [7].

From now on we will use the notation sApq instead L(p, q, s).

Theorem 4.4. Let 0 0. Define

M = sup0≤r<1

∫ 2π

0

|f(reiθ)|pdθ <∞ .

There exists 0 < r < 1 such that∫D\Dr

|f(w)|p dAq(w) =

∫ 1

r

(1 − ρ2)qρ

∫ 2π

0

|f(ρeiθ)|pdθ dr ≤ M

2

(1 − r2)q+1

q + 1<

ε

2.

Since (1− |ϕz(w)|2)s < 1, we decompose

lp,q,s(f)(z) ≤ maxw∈Dr

|f(w)|p∫Dr

(1 − |ϕz(w)|2)s dAq(z) +

∫D\Dr

|f(z)|p dAq(z) ,

then by Lemma 2.2 we conclude the proof by taking the limit when |z| → 1−. �

It is possible to replace the weight (1−|ϕz(w)|2)s by its reflection (|ϕz(w)|−2−1)s, as the following theorem shows.

Theorem 4.5. Let 0 0 and xs is nondecreasing for s > 0, we have in

general,∫D

|f(w)|p(1 − |ϕz(w)|2)s dAq(w) ≤∫D

|f(w)|p(|ϕz(w)|−2 − 1)s dAq(w) .

We claim that∫D

|f(w)|p(|φz(w)|−2 − 1)s dAq(w) ≤ t

∫D

|f(w)|p(1− |φz(w)|2)s dAq(w)

where t = t(q, s, R) is as in (4.2). If the result were not true, by change of variableformula with w = ϕz(v) and transposing terms,

0 <

∫D

|f(ϕz(v))|p(1− |ϕz(v)|2)q|ϕ′z(v)|2(|v|−2s − t) dAs(v)

≤ ρ(z, 2 + |q|)∫D

|f(ϕz(v))|p(1− |v|2)q(|v|−2s − t)dAs(v),

which leads to a contradiction with the inequality (4.2), since |f(ϕz(v))|p is asubharmonic function. �

5. Strict inclusions of the spaces A(p, q, s)

In this section we will prove the strict inclusions between weighted Bergman spacesfor 0 < s ≤ 1. For this, we need the following two lemmas. The first is due to A.Zygmund, (see [9]).

Lemma 5.1. Let 0 1, for all k ∈ N, then, there exists a constant A > 0,

depending only on p and λ, such that

1

A

( ∞∑k=1

|ak|2) 1

2

≤(

1

2π

∫ 2π

0

∣∣∣∣ ∞∑k=1

akeinkθ

∣∣∣∣p dθ) 1p

≤ A

( ∞∑k=1

|ak|2) 1

2

,

for any numbers ak, k ∈ N.

For n ∈ N, define

In = { k ∈ N : 2n ≤ k < 2n+1} .

The following lemma was proved by Mateljevic and Pavlovic in [3].

Lemma 5.2. Let 0 < α < ∞ and 0 0 depending only onp and α such that

1

K

∞∑n=0

tpn2nα

≤∫ 1

0

(1− x)α−1f(x)p dx ≤ K

∞∑n=0

tpn2nα

,

where tn =∑

k∈In ak.

Theorem 5.3. Let 0 < p < ∞, −2 < q < ∞ and 0 < s < 1 with −1 < q + s. Iff ∈ H has a power series expansion given by

f(w) =

∞∑n=0

anwn ,

and ∫ 1

0

( ∞∑n=1

|an|rn)p

(1− r2)q+s dr <∞ , (5.1)

then f ∈ s,0Apq.

Proof. Let 0 < R < 1. Consider the following estimation

lp,q,s(a) ≤ supw∈DR

|f(w)|p∫DR

(1 − |ϕz(w)|2)s dAq(w)

+

∫ 1

R

( ∞∑n=1

|an|rn)p

(1− r2)q+sr(1 − |a|2)s∫ 2π

0

dθ

|1− areiθ|2s dr.

We conclude the proof applying Lemmas 2.2, 2.1 and taking the limit when|z| → 1−. �

Corollary 5.4. Let 0 < p < ∞, −2 < q < ∞ and 0 < s ≤ 1, with −1 < q + s.Suppose that f ∈ H has a power series expansion given by

f(w) =∞∑n=0

anwn .

If∞∑n=0

1

2n(q+s+1)

(∑k∈In

|ak|)p

<∞

then f ∈ s,0Apq.

Proof. It is an immediate consequence of Theorem 5.3, condition (5.1) and Lemma5.2. �

Given f ∈ H with power series expansion

f(w) =

∞∑n=0

anwn

we say that f has Hadamard gaps of length λ > 1, if there exists an increasingsequence {nk} ⊂ N such that

an =

{0 si n = nk

anksi n = nk,

withnk+1

nk≥ λ > 1 , for k ∈ N.

We rewrite simply

f(w) =

∞∑k=0

akwnk .

Observe that the number of Taylor coefficients aj is at most [logλ 2] + 1 whennj ∈ Ik. The following theorem characterizes Lacunary series with Hadamardgaps in s,0Ap

q and sApq

Theorem 5.5. Let 0 1. Then the following statements are equivalent:

i) f ∈ s,0Apq ;

ii) f ∈ sApq;

iii) f ∈ Apq+s;

iv) the series∞∑k=0

1

2k(1+q+s)

( ∑nj∈Ik

|anj |2) p

2

(5.2)

is convergent.

Proof. By Corollary 3.11, it follows that i) implies ii). Now sApq ⊂ Ap

q+s, so ii)implies iii). We see now that iii) implies iv). By Lemmas 5.1 and 5.2, there existA > 0 and K > 0, such that∫

D

|f(w)|p dAq+s(w) =

∫D

∣∣∣∣ ∞∑k=0

akznk

∣∣∣∣p dAq+s(w)

=

∫ 1

0

∫ 2π

0

∣∣∣∣ ∞∑k=0

akrnkeinkθ

∣∣∣∣p(1 − r2)q+s r dθ dr

≥ 2π

Ap

∫ 1

0

( ∞∑k=0

|ak|2r2nk

) p2

r(1 − r2)q+s dr

≥ π

KAp

∞∑k=0

1

2k(q+s+1)

( ∑nj∈Ik

|aj |2) p

2

and the series (5.2) is convergent.We prove that iv) implies i). Since( ∑

nj∈In

|aj|)p

≤ 2p2 ([logλ 2] + 1)

p2

( ∑nj∈Ik

|aj |2) p

2

by Corollary 5.4 we have that f ∈ s,0Apq . �

Corollary 5.6. Let 0 < p < ∞, −2 < q < ∞ and 0 < s < 1 be with −1 < q + s.Then the inclusions

sApq ⊂

⋂s<t≤1

tApq and s,0Ap

q ⊂⋂

s<t≤1

t,0Apq

are strict.

Proof. Define f : D→ C by

f(w) =

∞∑n=0

2n(q+s+1)

p w2n .

Then, f ∈ H and for s < t ≤ 1,∞∑n=0

1

2n(1+q+t)(|an|2)

p2 =

∞∑n=0

2n(q+s+1)

2n(1+q+t)=

∞∑n=0

1

2(t−s)n<∞

and∞∑n=0

1

2n(1+q+s)(|an|2)

p2 =

∞∑n=0

2n(q+s+1)

2n(1+q+s)=

∞∑n=0

1 =∞ .

Therefore by Theorem 5.5

f ∈⋂

s<t≤1

t,0Apq but f /∈ s,0Ap

q . �

Theorem 5.7. Let 0 < p < ∞, −2 < q < ∞ and 0 < s < 1 with −1 < q + s. Ifp = 2k with k ∈ N then, f ∈ sAp

q if and only if

supz∈D

∞∑n=0

(1− |z|2)s(n+ 1)q+s+1

∣∣∣∣ n∑m=0

a{k}n Γ(n−m+ q + s)

(n−m)!zn−m

∣∣∣∣2 <∞ ,

where

a{k}n =1

2πi

∫ 2π

0

fk(ζ)

ζn+1dζ . (5.3)

Proof. As |f(w)|p = |fk(w)|2 with a{k}n given by (5.3), we have

fk(w)

(1− zw)s=

∞∑n=0

a{k}n wn∞∑

n=0

Γ(n+ s)

n!Γ(s)znwn

=

∞∑n=0

n∑m=0

a{k}m Γ(n−m+ s)

(n−m)!Γ(s)zn−mwmwn−m =

∞∑n=0

( n∑m=0

fn,m

)wn,

where,

fm,n = fa,m,n,k,s =Γ(n−m+ s)a

{k}m zn−m

(n−m)!Γ(s).

With this notation, we have∫D

|f(w)|p|1− zw|2s (1− |w|

2)s dAq(w) =

∫D

|fk(w)|2|1− zw|2s (1− |w|

2)s dAq(w)

=

∫D

∣∣∣∣ ∞∑n=0

( n∑m=0

fn,m)wn

∣∣∣∣2(1− |w|2)q+s dA(w)

=

∫ 1

0

∫ 2π

0

∣∣∣∣ ∞∑n=0

( n∑m=0

fn,m

)einθrn

∣∣∣∣2r(1 − r2)q+s dθ dr .

Next, we define

q(θ) =

∞∑n=0

( n∑m=0

fn,m

)einθ rn . (5.4)

We now calculate the Fourier coefficient of q(θ), that is,

〈q(θ), e−ikθ〉 =∞∑

n=0

( n∑m=0

fn,m

)∫ 2π

0

rn ei(n−k)θ dθ = 2π rkk∑

m=0

fk,m

= 2π rnn∑

m=0

fn,m .

By Parseval’s identity∫ 2π

0

∣∣∣∣ ∞∑n=0

( n∑m=0

fn,m

)einθ rn

∣∣∣∣2dθ =1

2π

∞∑n=0

∣∣∣∣ n∑m=0

fn,m

∣∣∣∣2 r2n .

Now, we have∫ 1

0

r2n+1(1− r2)q+sdr =1

2B(n+ 1, q + s+ 1) ≈ Γ(q + s+ 1)

2(n+ 1)q+s+1.

So finally, we get the theorem. �

Taking z = 0 in the previous theorem we obtain the following result.

Corollary 5.8. Let 0 < p < ∞, −2 < q < ∞ and 0 < s < 1 with −1 < q + s. Let

f ∈ H with power series given by

∞∑n=0

anwn. If p = 2k with k ∈ N, then f ∈ Ap

q+s

if and only if∞∑n=0

|an|2(n+ 1)q+s

<∞ .

6. F (p, q, s) and sApq spaces

For S ⊂ H, we define the primitive set of S as

P (S) = { h ∈ H : h′ ∈ S } .

Observe that if f ∈ sApq for 0 < p < ∞, −2 < q < ∞, 0 ≤ s < ∞, and h is a

primitive of f , then h ∈ F (p, q, s). Since by definition

supz∈D

∫D

|h′(z)|pgs(z, a) dAq(z) <∞ ,

that is, P (sApq) = F (p, q, s). Then by Theorem 4.2.1 of J. Rattya we have a

characterization of the sApq spaces in terms of higher derivatives with 1 < p <∞.

Theorem 6.1 ([5, Theorem 4.2.1]). Let f ∈ H and h a primitive of f . Let 1 < p <∞, −2 < q < ∞, 0 ≤ s < ∞ and n ∈ N be with −1 < q + s. Then, the followingconditions are equivalent:

a) h ∈ F (p, q, s);b) f ∈ sAp

q;

c) supa∈D

∫D

|f (n−1)(z)|p(1− |z|2)np−p(1− |ϕa(z)|2)s dAq(z) <∞ ;

d) supa∈D

∫D

|f (n−1)(ϕa(z))|p|ϕ′a(z)|np−p+q+2(1 − |z|2)np−p+s dAq(z) <∞ ;

e) supa∈D

∫D

|f (n−1)(z)|p(1− |z|2)np−pgs(z, a) dAq(z) <∞ .

By Theorem 4.2.2 of J. Rattya we have some unexpected relationships be-tween sAp

q spaces and F (p, q, s).

Theorem 6.2 ([5, Theorem 4.2.2]). Let h ∈ H. Let 1 < p < ∞, −2 < q < ∞ and0 ≤ s <∞ with −1 < q + s− p. Then the following conditions are equivalent:

a) h ∈ F (p, q, s);

b) supa∈D

∫D

|h(z)|p(1− |z|2)−p(1− |ϕa(z)|2)s dAq(z) <∞ ;

c) supa∈D

∫D

|h(z)|p(1− |z|2)−pgs(z, a) dAq(z) <∞ .

Corollary 6.3. For h ∈ H, let 1 < p < ∞, −2 < q < ∞ and 0 ≤ s < ∞ with−1 < q + s− p and −2 < q − p. Then F (p, q, s) =s Ap

q−p. In particular,

P (F (p, q, s)) = P (sApq−p) = F (p, q − p, s) .

7. Carleson measures

In this section we use Carleson type measures to characterize sApq .

For 0 < s < ∞, we say that a positive measure μ defined on D is a boundeds-Carleson measure provided

supI⊂∂D

μ(S(I))

|I|s <∞

where |I| denotes the arc length of I ⊂ ∂D and S(I) denotes the Carleson boxbased on I, that is

S(I) =

{w ∈ D :

w

|w| ∈ I, 1− |w| ≤ |I|2π

}.

If the next limit exists

lim|I|→0

μ(S(I))

|I|sfor I ⊂ ∂D, we say that μ is a compact s-Carleson measure. The following lemmacharacterizes s-Carleson measures (see Lemma 2.2 from [1]).

Lemma 7.1. Let 0 < s <∞ and μ be a positive measure on D. Then

i) μ is a bounded s-Carleson measure if and only if

supz∈D

∫D

|ϕ′z(w)|s dμ(w) <∞ ;

ii) μ is a compact s-Carleson measure if and only if

lim|z|→1

∫D

|ϕ′z(w)|s dμ(w) = 0 .

The following result is immediate.

Theorem 7.2. Let 0 < s <∞, 0 < p <∞ and −2 < q <∞. Consider f ∈ H anddμf (w) = |f(w)|p(1− |w|2)q+s dA(w). Then,

i) f ∈ sApq if and only if dμf (w) is a bounded s-Carleson measure.

ii) f ∈ s,0Apq if and only if dμf (w) is a compact s-Carleson measure.


Acknowledgment

Many thanks to the Referee.

References

[1] R. Aulaskari, D. Stegenga and J. Xiao, Some subclasses of BMOA and their char-acterization in terms of Carleson measures, Rocky Mountain J. Math. 26, (1996),485–506.

[2] E. Hewitt, K. Stromberg Real and Abstract Analysis, Springer-Verlag, 1975.

[3] M. Mateljevic, M. Pavlovic, Lp behaviour of power series with positive coefficientsand Hardy spaces, Proc. Amer. Math. Soc. 87, (1983), 309–316.

[4] J. Perez H., L.F. Resendis O. and L.M. Tovar S., Quaternionic Bergman Spaces,Preprint (2014), 1–12.

[5] J. Rattya, On some complex Function Spaces and Classes, Ann. Acad. Scie. Fenn.Math. Diss. 124, (2001), 1–73.

[6] R. Zhao, On α-Bloch functions and VMOA, Acta Mathematica Scientia, 16 (3),(1996), 349–360.

[7] R. Zhao, On a general family of function spaces, Ann. Acad. Scie. Fenn. Math. Diss.105, (1996), 1–56.

[8] K. Zhu, Operator Theory in Function Spaces, Marcel Dekker, New York, 1990.

[9] A. Zygmund, Trigonometric Series. Cambridge, Univ. Press, London and New York,1959.

Luıs Javier Carmona L.Universidad Autonoma MetropolitanaUnidad Iztapalapa, Departamento de MatematicasAv. San Rafael Atlixco Num.186C.P. 09340, Mexico D.F.

e-mail: carmona [email protected]

Lino Feliciano Resendis OcampoUniversidad Autonoma MetropolitanaUnidad Azcapotzalco, Departamento de Ciencia BasicasAv. San Pablo 180. Col. Reynosa TamaulipasC.P. 02200, Mexico D.F.


Luis Manuel Tovar SanchezEscuela Superior de Fısica y Matematicas del IPNEdif. 9, Unidad ALM,Zacatenco del IPN.C.P. 07300, Mexico D.F.


mailto:carmona [email protected]





On Appell Sets and Verma Modules for sl(2)

David Eelbode and Nikolaas Verhulst

Abstract. The aim of this paper is to introduce a general framework whichcan be used to generalize both Appell sets in multivariate analysis and specialpolynomials in a complex variable z ∈ C, inspired by certain special functionsappearing in harmonic and Clifford analysis on Rm. As an illustration, wehave a closer look at Hermite polynomials.

Mathematics Subject Classification (2010). 30G35; 17B10; 33C45.

Keywords. Appell sets, Verma modules, Hermite polynomials.

1. Introduction

Mathematical analysis and classical representation theory have always gained froma mutual infusion of ideas and techniques. The former often provides concrete ex-amples which may illustrate and initiate more abstract notions which are studiedand generalized in the latter. The present paper is written with this philosophyin mind, hereby drawing inspiration from two particular problems arising in har-monic analysis on Rm. First of all, there exist quite a few function theories whichare centered around a set of operators which realize a copy of the simple Lie al-gebra sl(2) under the commutator bracket. Classical harmonic analysis itself, forexample, is centered around the Laplace operator Δm, acting as an endomorphismon C[x1, . . . , xm]. The associated Lie algebra sl(2) is then given by

sl(2) ∼= Alg(X,Y,H

) ∼= Alg

(1

2Δm,−1

2|x|2,−Ex − m

2

), (1.1)

where |x|2 is the squared norm of x ∈ Rm and Ex =∑

xj∂xj stands for the Euleroperator on Rm. A celebrated result in harmonic analysis, due to R. Howe [10],describes this Lie algebra as the so-called dual partner of the orthogonal groupSO(m), acting on polynomials in C[x1, . . . , xm] through the regular representa-tion, leading to a multiplicity-free decomposition of this space in terms of Vermamodules for sl(2). Also in super analysis (a function theory in which both com-muting and anti-commuting variables are taken into account, see, e.g., [5]) andthe theory of Dunkl operators (in which the rotational symmetry is reduced to a

Switzerland

112 D. Eelbode and N. Verhulst

finite subgroup, see, e.g., [6, 7]), a key ingredient is the existence of a subalgebraA ∼= sl(2) inside the full endomorphism algebra acting on the polynomials (orsmooth functions in general). This algebra is then used to define generalizationsof the classical Hermite polynomials, which lead to an expression for the Fourierintegral as an exponential operator exp(A), with A ∈ A.

A second motivation comes from the theory on Appell sequences. Classi-cally, these are defined in terms of a complex variable z, as sets of holomor-phic polynomials {Pk(z) : k ∈ Z+} satisfying the relation P ′

k(z) := kPk−1(z),where degPk(z) = k and P0 = 0. Denoting the derivation operator by means ofP = ∂z, one can also interpret these Appell sequences as representations for theHeisenberg–Weyl algebra h1, provided there exists an associated raising operatorM satisfying MPk(z) = Pk+1(z) for all k ∈ Z+. Formally, this operator thusperforms the integration within the Appell sequence. One can indeed verify thatthese defining relations imply that P and M satisfy the canonical commutationrelation [P,M ] = 1 ∈ C = z(h1). Notable examples are the Hermite polynomials,but also other orthogonal polynomials such as Bernoulli and Euler polynomials.More generally, we say:

Definition 1.1. A representation for the Heisenberg algebra (denoted by meansof h1) is an algebra morphism ρV : h1 → End(V). If there exists a vector v0 inker ρV (P ), we will from now on refer to the set A := {ρkV (M)[1] : k ∈ Z+} as anAppell sequence related to the operators (ρV (P ), ρV (M)).

In the classical situation, the vector space V = span(Pk(z))k is a polynomialfamily, with ρV (P ) a differential operator and ρV (M) an conjugation thereof (aformal integration operator). In this paper, we will consider generalizations of Ap-pell sequences, for which V may be a more abstract representation space. Note thatthe representation V must be infinite dimensional, which follows from the fact thattr([ρV (P ), ρV (M)]) = 0, as it is the trace of a commutator. This means that finite-dimensional representations must necessarily satisfy dim(V) = 1. This suggestslooking for generalizations of Appell sequences in canonical infinite-dimensionalmodules for sl(2), i.e., Verma modules.

Recently, the problem of constructing analogues of complex Appell sequencesin multivariate analysis has gained new interest, see a.o. [3, 9, 11, 8, 2]. These se-quences are defined as polynomial sets V = span(Pk(x))k containing scalar-valued(resp. Clifford algebra-valued) null solutions for the Laplace or Dirac operator inthe variable x ∈ Rm, for which the lowering operator P is a differential operatorbelonging to the Clifford–Weyl algebraWC

m, defined by WCm := Alg

(xi; ∂xj

)⊗Cm

(with Cm the universal Clifford algebra in m dimensions). It turns out that someof these Appell sequences can be related to the branching problem for certainirreducible representations for the spin group, the (inductive) construction of or-thonormal bases for these spaces and generalizations of the classical Fueter theo-rem, see, e.g., [1, 8]. This has given rise to Gegenbauer and Jacobi polynomials inharmonic (resp. Clifford) analysis, for which the formal integration operator wasobtained in terms of an operator containing fractions.

Appell Sets and Verma Modules 113

2. Appell sets in Verma modules for sl(2)

Adopting the classical definition (which can be given for any Lie algebra g) to ourcase of interest, we have the following:

Definition 2.1. Let λ ∈ C be a complex number, and consider the vector spaceVλ = Cvλ on which sl(2) acts as H [vλ] = λvλ and X [vλ] = 0, Y [vλ] = 0. Onemay then define the highest-weight Verma module Mλ := U(n−)⊗U(b+) Vλ, where

n− = CY and b+ = CH ⊕ CX , with highest weight λ ∈ C.

In view of the PBW-theorem for the universal enveloping algebra U(sl(2)),the weight space decomposition is given by Mλ = spanC

(Y k ⊗ vλ : k ∈ Z+

). The

following fact about Verma modules is well known:

Proposition 2.2. The Verma modules Mλ are irreducible for λ /∈ Z+.

Remark 2.3. In this paper, we have chosen to work with highest-weight Vermamodules Mλ, but the construction for lowest-weight modules is similar.

The conditions on λ ensuring that Mλ is irreducible are very convenient, inthe sense that we will only be able to introduce generalizations of Appell sequencesas algebra morphisms σλ from h1 into End(Mλ) under the same conditions on λas in the proposition above (which means that the action of σλ(P ) and σλ(M)will only be well defined for λ ∈ C \Z+). In what follows, we consider the subringR := U(b−) ⊂ U(sl(2)), where we use the notation b− := CH⊕CY for the (other)Borel subalgebra (compare with b+).

Remark 2.4. Fractions will play a crucial role throughout this paper, and wetherefore define them as A/B := AB−1, where the inversion always appears at theright-hand side. This is important, because in general A and B will belong to anon-commutative ring.

Definition 2.5. For all α ∈ C, we define S−α :=

⟨(H + α + 2j) : j ∈ Z−⟩ ⊂ R, as

the set which is multiplicatively generated by the elements between brackets.

It is then easily verified that S−α ⊂ R satisfies the right Øre condition for

arbitrary α ∈ C, see, e.g., [4]. This condition is needed whenever one wants toconsider the right ring of fractions R(S−1) with respect to a multiplicatively closedsubset S ⊂ R. Indeed, for arbitrary ξ ∈ R and σ ∈ S−

α one has that ξS−α ∩ σR =

∅. Indeed, in view of the PBW-theorem for U(sl(2)), it suffices to consider an

element ξ of the form ξ = Y aHb (hereby omitting the tensor product symbolsand with a, b ∈ Z+). For σ = H + α + 2j (with j ∈ Z−), one then clearly hasthat (H + α+ 2j)Y aHb = Y aHb

(H + α+ 2(j − a)

), since 2(j − a) ∈ Z−. We can

thus define the localization w.r.t. the set S−α , which will be denoted by means of

R−α := R

((S−

α )−1).

Remark 2.6. Since (H + α + 2j)Y aHbXc = Y aHbXc(H + α + 2(j + c − a)

), it

is clear that one can also consider the localization of the full enveloping algebraU(sl(2)) with respect to the (enlarged) subset Sα :=

⟨(H + α + 2j) : j ∈ Z

⟩,


where j ∈ Z is now an arbitrary integer. We therefore also introduce the notationUα(sl(2)

):= U(sl(2))(S−1

α ). Obviously, for all α ∈ C one has that R−α ⊂ Uα

(sl(2)

).

Note that these are all inside the skew field over U(sl(2)).The motivation for considering this particular localization R−

α , instead ofUα(sl(2)

), comes from the fact that the action of the latter localization will not

always be well defined on the Verma modules we would like to consider for practicalpurposes. In full generality, one has the following:

Proposition 2.7. Whenever (λ + α) /∈ 2Z+, the action of R−α is well defined on

irreducible Verma modules Mλ.

Proof. To prove that the action on the localization w.r.t. S−α is well defined, we

first of all note that the elements in S−α act as a constant on the weight spaces.

It then suffices to verify that for all integers k ≥ 0 and j ≤ 0 one has that(H +α+ 2j)[Y k ⊗ vλ] = (α+ λ− 2k+2j)Y k ⊗ vλ = 0. This is indeed guaranteedwhenever α+ λ = 2(k + |j|) ∈ 2Z+. �Corollary 2.8. The action of R−

λ is well defined on irreducible Verma modules Mλ

(i.e., for arbitrary λ ∈ C such that λ is not a positive integer).

Corollary 2.9. For arbitrary α ∈ C, the action of Uα(sl(2)

)is well defined on

irreducible Verma modules Mλ whenever α+ λ /∈ 2Z.

Remark 2.10. Note that the action of Uλ(sl(2)

)on Mλ is not always well defined,

in view of the fact that, e.g., for λ = l ∈ Z− and j = k − l we get that (H + λ +2j)[Y k ⊗ vλ] = 0.

Let us then prove the main result of this section:

Theorem 2.11. Suppose Mλ is an irreducible Verma module for the algebra sl(2),which means that λ /∈ Z+. One can then define an action of h1 on Mλ, by meansof the algebra morphism

σλ : h1 → End(Mλ) : (P,M) �→(X,

2Y

H + λ

).

Note that the operator σλ(M) actually belongs to the localization R−λ , which means

that its (repeated) action on Mλ is well defined (see Corollary 2.8).

Remark 2.12. Recalling the notation from the introduction, we thus have that thebasis for Mλ defined by A := {σk

λ(M)[1⊗vλ] : k ∈ Z+} defines an Appell sequence.

Proof. It suffices to verify that the action of σλ(P ) and σλ(M) on the module Mλ

satisfies the Heisenberg relation [σλ(P ), σλ(M)] = 1. For that purpose, we notethat for all k > 0, we have:

[σλ(P ), σλ(M)](Y k ⊗ vλ) =

(XY

2

H + λ− Y

2

H + λX

)(Y k ⊗ vλ)

= Y k ⊗ vλ .

For k = 0, the statement is trivial, which proves the theorem. �


Invoking the definition (λ)k = λ(λ − 1) · · · (λ − k + 1), we then have:

Definition 2.13. Suppose Mλ is a highest-weight Verma module for sl(2), withλ ∈ C \ Z+. The monomial basis for Mλ is given by the weight vectors

vλ(k) :=1

(λ)kY k ⊗ vλ (k ∈ Z+) .

Note that the embedding of h1 into Uλ(sl(2)

)also gives rise to another Appell

sequence. To see this, we will calculate the commutator [σλ(P ), σλ(M)] inside thelocalization and then investigate its action on Verma modules Mμ (with μ ∈ C

arbitrary). In view of the fact that (H + λ)X = X(H + λ+ 2), we get:

[σλ(P ), σλ(M)] =2H

H + λ+ 2Y

[X,

1

H + λ

]= 2

H(H + λ+ 2) + 2Y X

(H + λ)(H + λ+ 2).

When acting on an arbitrary weight space in the module Mμ, we thus get:

[σλ(P ), σλ(M)]Y k ⊗ vμ = 2(μ− 2k)(μ+ λ− 2k + 2) + 2k(μ+ 1− k)

(μ+ λ− 2k)(μ+ λ− 2k + 2)Y k ⊗ vμ .

It is then easily verified that the Appell condition is verified (i.e., the constantin front of the weight vector is equal to 1) for μ ∈ {λ, 2− λ}. This means that wehave now obtained the following (somewhat stronger) result:

Corollary 2.14. Consider a complex number λ ∈ C \ Z. One can then define

σλ : h1 → End(Mμ) : (P,M) �→(X,

2Y

H + λ

),

for μ ∈ {λ, 2 − λ}. Both Verma modules Mλ and M2−λ then become Appell se-quences for the operators σλ(P ) and σλ(M). Note that the latter operator belongsto R−

λ , which means that its action is always well defined.

Note that we imposed the condition λ /∈ Z in the corollary above, to ensure thatboth λ, μ = 2− λ ∈ C \ Z+. Let us then consider a few examples:

(i) Consider the classical realization for sl(2) in harmonic analysis on Rm, see(1.1). It is then clear that we can start from an arbitrary harmonic functionfα(x) on an open subset Ω ⊂ Rm which is homogeneous of degree α ∈ C.This function then plays the role of a highest weight vector for which λ =−α− m

2 /∈ Z−, leading to the Appell sequence

Aλ :=

{|x|2kfα(x)

2k(α+ m

2

) (α+ m

2 + 1) · · · (α+ m

2 + k − 1)} ,

for the lowering operator P = 12Δx. In case α+ m

2 /∈ Z, we can also consideran Appell sequence starting from Mμ, with μ = 2 − λ. In the context ofharmonic analysis, there is a well-known realization for the highest weightvector for Mλ in terms of the Kelvin inversion:

J0 : fα(x) �→ 1

|x|m−2f

(x

|x|2)

= |x|2−m−2αfα(x) .


This gives rise to the Appell sequence

Aμ :=

{(−1)k|x|2−m−2α+2kfα(x)

2k(α+ m

2 − 2) (

α+ m2 − 3

) · · · (α+ m2 − k − 1

)} .

(ii) In [8], we have obtained the harmonic (resp. monogenic) Gegenbauer polyno-mials through the knowledge of a particular subalgebra of the Weyl algebraWm (resp. WC

m), for all m ≥ 3 given by

sl(2) ∼= Alg(− ∂xm , xm(2Ex +m− 2)− r2∂xm ,−2Ex − (m− 2)

).

It is then clear that the polynomial set

G2−m :={(xm(2Ex +m− 2)− r2∂xm)k[1] : k ∈ N

}can be considered as a highest-weight Verma module Mλ with highest weightvector 1 ∈ C, for λ = −(m− 2).

3. Hermite bases in Verma modules for sl(2)

One can now develop a general framework to define special polynomials (e.g.,Hermite polynomials). Traditionally, such polynomials can be defined through an

explicit formula of the form Sk(z) =∑k

j=0 cj,k(S)zj , with z ∈ C and cj,k(S) a

certain coefficient that determines the special function under consideration. Wewill generalize this picture, hereby using the following idea: instead of using acomplex variable z, we will use the operator σλ(Y ) which creates the monomialbasis for an arbitrary (fixed) Verma module Mλ, hereby fixing the realizationsl(2) = Alg(X,Y,H). For example, the Hermite basis for the Verma module Mλ isthen defined through the repeated action of the following operators in Uλ

(sl(2)

):

σ(h)λ (P ) := X and σ

(h)λ (M) := σλ(M − P ) =

2Y

H + λ−X .

Note that we have added a superscript (h) to indicate that these generate theHermite basis, corresponding to the probabilists’ Hermite polynomials, as opposedto the physicists’ Hermite polynomials which would require adding a factor 2 tothe term σλ(M). The raising operator can also be defined as

σ(h)λ (M) = − exp

(1

2σ2λ(M)

)σλ(P ) exp

(−1

2σ2λ(M)

), (3.1)

where the exponential is defined through its formal Taylor expansion.

Definition 3.1. Suppose Mλ is a Verma module, with λ ∈ C\Z+ and weight spacesY k ⊗ vλ (k ∈ Z+). The Hermite basis for Mλ is then given by the following set ofvectors:

v(h)λ (k) :=

(2Y

H + λ−X

)k

[1⊗ vλ] (k ∈ Z+) .


As a result of expression (3.1), this can also be written as follows:

v(h)λ (k) = (−1)k exp

(1

2σ2λ(M)

)σkλ(P ) exp

(−1

2σ2λ(M)

)[1⊗ vλ] .

The fact that this defines a basis follows from the following proposition, the essenceof which is encoded in a technical lemma:

Lemma 3.2. For all k ∈ Z+, we have the following expansion of binomial typewhen acting on the highest weight vector 1⊗ vλ:(

σλ(M)− σλ(P ))2k

=

k∑j=0

(−1)jk!2jj!(k − 2j)!

σ2k−2jλ (M)

(σλ(M)− σλ(P )

)2k+1=

k∑j=0

(−1)jk!2jj!(k − 2j)!

σ1+2k−2jλ (M).

Proof. The theorem can be easily proved by induction, taking into account thatfactors σλ(P ) may safely be ignored once they are at the right-hand side (inview of the fact that the expression is meant to act on the highest weight vector1⊗ vλ ∈Mλ). �

Proposition 3.3. The explicit expression for the Hermite basis vectors for a Vermamodule Mλ in terms of the monomial basis, is given by:

v(h)λ (k) =

κ∑j=0

(−1)jk!2jj!(k − 2j)!(λ)k−2j

Y k−2j ⊗ vλ ,

hereby introducing the integer κ := �k2� ∈ Z+.

Proof. This immediately follows from the previous lemma, hereby making use ofthe fact that σλ(M) generates the monomial basis for Mλ. �

The classical Hermite polynomial Hk(x) in a real variable x ∈ R, as in theprobabilistic normalization, corresponds to the case where monomial basis vectorsvλ(k) ∈Mλ are identified with monomials xk. Note that the Hermite basis vectorssatisfy the following recurrence relations:

v(h)λ (k + 1) =

1

(λ− k)Y v

(h)λ (k)−Xv

(h)λ (k)

=1

(λ− k)Y v

(h)λ (k)− kv

(h)λ (k − 1) ,

where we explicitly made use of the fact that the Hermite basis vectors define anAppell sequence. This gives then rise to the following eigenvalue problem for thelinear operator Lλ ∈ Uλ

(sl(2)

), which is the equivalent of the Hermite equation in

the classical context:

Lλv(h)λ (k) :=

(X2 − 2Y

H + λ

)v(h)λ (k) = −kv(h)λ (k) .


References

[1] S. Bock, K. Gurlebeck, R. Lavicka, V. Soucek, V., The Gelfand–Tsetlin bases forspherical monogenics in dimension 3, Rev. Mat. Iberoam. 28, Issue 4 (2012), 1165–1192.

[2] I. Cacao, D. Eelbode, Jacobi polynomials and generalized Clifford algebra-valued Ap-pell sequences, to appear in Math. Meth. Appl. Sc., doi: 10.1002/mma.2914.

[3] I. Cacao, I. Falcao, H. Malonek, Laguerre derivative and monogenic Laguerre poly-nomials: an operational approach, Math. Comput. Model. 53 (2011), 1084–1094.

[4] P.M. Cohn, Skew fields, Theory of general division rings, Cambridge UniversityPress, 1995.

[5] H. De Bie, Fourier transform and related integral transforms in superspace, J. Math.Anal. Appl. 345 (2008), 147–164.

[6] H. De Bie, An alternative definition of the Hermite polynomials related to the DunklLaplacian, SIGMA 4 (2008), 093, 11 pages.

[7] C.F. Dunkl, Differential-difference operators associated to reflection groups, Trans.Am. Math. Soc. 311 (1989), 167–183.

[8] D. Eelbode, Monogenic Appell sequences as representations of the Heisenberg algebra,Adv. Appl. Cliff. Alg. 22, Issue 4 (2012), 1009–1023.

[9] I. Falcao, H. Malonek, Generalized exponentials through Appell sets in Rn+1 andBessel functions in: Simos, T.E., Psihoyios, G., Tsitouras, C. (eds.), AIP ConferenceProceedings, Vol. 936 (2007), 738–741.

[10] R. Howe, Remarks on classical invariant theory, Trans. Amer. Math. Soc. 313 No. 2(1989), 539–570.

[11] Malonek, H.R., Tomaz, G., Bernoulli polynomials and Pascal matrices in the contextof Clifford analysis, Discr. Appl. Math. 157 No. 4 (2009), pp. 838–847.

David Eelbode and Nikolaas VerhulstUniversity of Antwerp Campus Middelheim (Building G)Middelheimlaan 1B-2020 Antwerp, Belgiume-mail: [email protected]

[email protected]





Integral Formulas for k-hypermonogenicFunctions in R3

Sirkka-Liisa Eriksson, Heikki Orelma and Nelson Vieira

Abstract. We consider harmonic functions with respect to the Laplace–Bel-trami operator of the Riemannian metric ds2 = x−2k

2

(∑2i=0 dx

2i

)and their

quaternion function theory in R3. Leutwiler noticed around 1990 that if theusual Euclidean metric is changed to the hyperbolic one, that is k = 1,then the power function (x0 + x1e1 + x2e2)

n, calculated using quaternions,is the conjugate gradient of a hyperbolic harmonic function. We study gen-eralized holomorphic functions, called k-hypermonogenic functions satisfyingthe modified Dirac equation. Note that 0-hypermonogenic are monogenic and1-hypermonogenic functions are hypermonogenic defined by H. Leutwiler andthe first author.

We prove the Cauchy type integral formulas for k-hypermonogenic wherethe kernels are calculated using the hyperbolic distance of the Poincare upperhalf-space model. Earlier these results have been proved for hypermonogenicfunctions.

Mathematics Subject Classification (2010). Primary 30A05; Secondary 30A45.

Keywords. Hypermonogenic, hyperbolic, Laplace–Beltrami, monogenic, qua-ternions.

1. Introduction

We study generalized function theory connected to the hyperbolic metric. In thisframe work the generalized holomorphic functions are called k-hypermonogenicfunctions introduced in [1]. The theory of k-hypermonogenic functions is con-necting the theory of monogenic functions with the value k = 0 and theory ofhypermonogenic functions with value k = n − 1. Moreover, it also connected tothe eigenfunctions of Laplace–Beltrami operator with respect to the hyperbolicmetric of the Poincare upper half-space model.

Hypermonogenic functions were introduced in Rn+1 by H. Leutwiler and thefirst author in [6]. An introduction to the theory is presented in [8] or in [10].

Switzerland

120 S.-L. Eriksson, H. Orelma and N. Vieira

The first Cauchy type integral formulas for them were proved in [7] and the totalformula with two kernels in [2]. They were improved to contain just one singlekernel in [9] and [4]. Later in [5] it was noticed a surprising result that the kernelis the Cauchy kernel of monogenic function shifted to the Euclidean center of thehyperbolic ball.

The general Cauchy type theorem for k-hypermonegenic functions was provedin [3] but the kernels have a complicated formula. In the main result of this paperwe present a different way to compute the kernels and an explicit integral formulain R3. In consecutive papers the integral operators will be studied. The resultsmay be also generalized to higher dimensions.

2. Preliminaries

We review the main notations and concepts. Let H be the real associative divisionalgebra of quaternions generated by e1, e2, e3 satisfying the properties e1e2 =e3 and eiej + ejei = −2δij where δij is the usual Kronecker delta. Elementsx = x0 + x1e1 + x2e2 for x0, x1, x2 ∈ R are called paravectors. The vector spaceR3 is identified with the real vector space of paravectors and therefore elementsx0 + x1e1 + x2e2 and (x0, x1, x2) are identified. The field of complex numbers isidentified with the set {x0 + x1e1| x0, x1 ∈ R}.

We use several involutions. Let q = q0+q1e1+q2e2+q3e3. The main involutionis the mapping q → q′ defined by

q′ = q0 − q1e1 − q2e2 + q3e3.

Similarly the reversion is the anti-involution q → q∗ defined by

q′ = q0 + q1e1 + q2e2 − q3e3.

The conjugation is the anti-involution q → q that is the composition of the pre-

ceding involutions, that is q = (q′)∗ = (q∗)′and

q = q0 − q1e1 − q2e2 − q3e3.

The following product rules hold

(ab)′ = a′b′ (ab)∗ = b∗a∗ ab = b a

for all quaternions a and b. Using the unique decomposition q = u+ve2 for u, v ∈ C

we define the mappings P : H → C and Q : H → C by Pq = u and Qq = v (see[6]). In order to compute the P - and Q-parts, we define the involution q → q by

q = q0 + q1e1 − q2e2 − q3e3.

Then we obtain the formulas

Pq =1

2(q + q) , (2.1)

Qq = −1

2(q − q) e2. (2.2)

Integral Formulas 121

Note that the product rule ab = ab holds for all quaternions a and b. The followingproduct rules ([6]) hold

P (ab) = (Pa)Pb+ (Qa)Q (b′) , (2.3)

Q (ab) = (Pa)Qb+ (Qa)P ′ (b) (2.4)

= aQb+ (Qa) b′.

Note that if a ∈ H then

a′e2 = e2a.

Moreover, if a ∈ C then

ae2 = e2a′. (2.5)

Our general assumption is that a function f (x0, x1, x2) : Ω → H is definedon an open subset Ω of R3 and its components are continuously differentiable. Theleft Dirac operator and the right Dirac operator in H are defined by

Dlf =∂f

∂x0+ e1

∂f

∂x1+ e2

∂f

∂x2, Drf =

∂f

∂x0+

∂f

∂x1e1 +

∂f

∂x2e2.

Their conjugate operators Dl and Dr are introduced by

Dlf =∂f

∂x0− e1

∂f

∂x1− e2

∂f

∂x2, Drf =

∂f

∂x0− ∂f

∂x1e1 − ∂f

∂x2e2.

These operators are also called generalized Cauchy–Riemann operators.

The modified Dirac operatorsM lk, M

l

k, Mrk and M

r

k are introduced in ([1]) by

M lkf (x) = Dlf (x) + k

Qf (x)

x2, M r

kf (x) = Drf (x) + kQf (x)

x2,

Ml

kf (x) = Dlf (x)− kQf (x)

x2, M

r

kf (x) = Drf (x) + kQf (x)

x2

for x ∈ {x ∈ Ω | x2 = 0}. Since the operatorM l1 is directly connected to the hyper-

bolic Poincare upper half-space model it is specially important and is also denotedbriefly by M (introduced in ([6]).

Definition 2.1. Let Ω ⊂ R3 be an open set. Let k ∈ R. A mapping f : Ω → H

is called left k-hypermonogenic, if f ∈ C1 (Ω) and M lkf (x) = 0 for any x ∈

{x ∈ Ω | x2 = 0}. The 0-hypermonogenic functions are called monogenic. The 1-hypermonogenic functions are called briefly hypermonogenic. The right k-hyper-monogenic functions are defined similarly. A twice continuously differentiable func-

tion f : Ω → H is called k-hyperbolic harmonic if Ml

kMlkf = 0 for any x ∈

{x ∈ Ω | x2 = 0} .


3. Integral formulas for k-hypermonogenic functions

The main idea is that we use the hyperbolic metric

ds2 =dx2

0 + dx21 + dx2

2

x22

(3.1)

of the Poincare upper half-space model instead of the Euclidean one. Note thatthe operator

Δhf = x22Δf − x2

∂f

∂x2

is the hyperbolic Laplace–Beltrami operator with respect to this metric. The hy-perbolic distance may be computed as follows (see [14]).

Lemma 3.1. The hyperbolic distance dh(x, a) with respect to the metric (3.1) be-tween the points x and a in R3

+ is

dh(x, a) = arcosh λ(x, a),

where

λ(x, a) =|x− a|2 + |x− a|2

4x2a2=|x− a|22x2a2

+ 1

and |x− a| is the usual Euclidean distance between the points a and x.

We note also the relation between the Euclidean and hyperbolic balls.

Proposition 3.2. The hyperbolic ball Bh (a, rh) in R3+ with the hyperbolic center

a = a0 + a1e1 + a2e2 and the radius rh is the same as the Euclidean ball with theEuclidean center

ca (rh) = a0 + a1e1 + a2 cosh rhe2

and the Euclidean radius re = a2 sinh rh.

There are two main relations between k-hyperbolic harmonic functions andk-hypermonogenic functions recalled as follows.

Theorem 3.3 ([6]). Let Ω ⊂ R3 be an open set and f : Ω→ H be twice continuouslydifferentiable. Then f is k-hypermonogenic if and only if f and xf (x) are k-hyperbolic harmonic functions.

Proposition 3.4 ([6]). Let Ω ⊂ R3 be an open set. Let k ∈ R. If a mapping h :

Ω → H is k-hyperbolic harmonic then Ml

kh is k-hypermonogenic. Conversely, ifa mapping f : Ω → H is k-hypermonogenic there exists locally a complex k-hyperbolic harmonic function h satisfying f = Dh.

We use the following calculation rules, proved in [11]

Lemma 3.5. If c (x, a) = Pa+ a2 coshdh (x, a) e2 then

x2Dxλ (x, a) =

x− c (x, a)

a2,

x2Dxdh (x, a) =

x− c (x, a)

a2 sinh dh (x, a)=

x− c (x, a)

|x− c (x, a)| .


Our key tool is the relation between k-hyperbolic harmonic functions and theeigenfunctions of the hyperbolic Laplace–Beltrami operator, stated slightly moregeneral form next.

Proposition 3.6 ([12]). If u is a solution of the equation

x22 � u (x)− kx2

∂u

∂x2(x) + lu (x) = 0 (3.2)

in an open subset Ω ⊂ R3+, then f(x) = x

1−k2

2 u(x) is an eigenfunction of the

hyperbolic Laplace operator corresponding to the eigenvalue 14

(k2 + 2k − 3− 4l

).

Conversely, if f is an eigenfunction of the hyperbolic Laplace operator correspond-

ing to the eigenvalue γ in an open subset Ω ⊂ R3+ then u(x) = x

k−12

2 f(x) is the

solution of the equation (3.2) in Ω with γ = 14

(k2 + 2k − 3− 4l

).

We are looking for eigenfunctions of the hyperbolic Laplace depending juston the hyperbolic distance dh (x, e2). We first recall a formula of the hyperbolicLaplace for functions depending only on dh (x, e2) .

Lemma 3.7 ([13]). If f is twice continuously differentiable depending only on rh =dh (x, e2) then the hyperbolic Laplace in R3

+ is given by

�hf (rh) =∂2f

∂r2h+ 2 coth rh

∂f

∂rh.

Eigenfunctions of the hyperbolic Laplace depending on rh are related to aneasier differential equation.

Lemma 3.8. If f is a solution of the equation

∂2f

∂r2h+ γf = 0

then the function g (rh) =1

sinh rh

∂f∂rh

, depending only on rh, is an eigenfunction of

the hyperbolic Laplace operator corresponding to the eigenvalue − (γ + 1) .

Proof. We just compute

∂2g

∂r2h=

(− 1

sinh rh+ 2

cosh2 rh

sinh3 rh

)∂f

∂rh− 2

cosh rh

sinh2 rh

∂2f

∂r2h+

1

sinh rh

∂3f

∂r3h,

2∂g

∂rh

cosh rhsinh rh

= −2 cosh2 rh

sinh3 rh

∂f

∂rh+

2 cosh rh

sinh2 rh

∂2f

∂2rh.

Since ∂2f∂r2h

= −γf we conclude that 1sinh rh

∂3f∂r3h

= g and

∂2g

∂r2h+ 2

∂g

∂rh

cosh rhsinh rh

=1

sinh rh

∂3f

∂r3h− g = − (γ + 1) g. �

Applying the previous lemma we obtain the general solution depending onthe hyperbolic distance.

Theorem 3.9. The general solution of the equation

∂2f

∂r2h+ 2 coth rh

∂f

∂rh+ γf = 0 (3.3)

is

f (rh) =

⎧⎪⎪⎨⎪⎪⎩C cosh(

√1−γrh)

sinh rh+

C0 sinh(√1−γrh)

sinh rh, if γ < 1,

Csinh rh

+ C0rhsinh rh

, if γ = 1,

C cos(√γ−1rh)

sinh rh+

C0 sin(√γ−1rh)

sinh rh, if γ > 1,

for some real constants C and C0.

Corollary 3.10. The bounded solutions of (3.3) are

f (rh) =

⎧⎪⎪⎨⎪⎪⎩C sinh(

√1−γrh)

sinh rh, if γ < 1,

Crhsinh rh

, if γ = 1,

C sin(√γ−1rh)

sinh rh, if γ > 1.

Corollary 3.11. The particular solution of (3.3) with a singularity at e2 is

f (rh) =

⎧⎪⎪⎨⎪⎪⎩cosh(

√1−γrh)

sinh rh, if γ < 1,

1sinh rh

, if γ = 1,

cos(√γ−1rh)

sinh rh, if γ > 1.

Corollary 3.12. The particular solution of (3.3) with γ = 14 (4 − (k + 1)2) outside

the point e2 is

F (x) =cosh

(dh(x,e2)(k+1)

2

)sinh dh (x, e2)

=cosh

(dh(x,e2)(k+1)

2

)|x− cosh dh (x, e2)|

and xk−12

2 F (x) is k-hyperbolic harmonic.

If we transform the preceding function with the hyperbolic translation τ (x) =a2x+ Pa, we note that

F (x) =cosh

(dh(τ(x),a)(k+1)

2

)sinh dh (τ (x) , a)

and

F(τ−1 (u)

)=

cosh(

dh(u,a)(k+1)2

)sinh dh (u, a)

is also an eigenfunction of the hyperbolic Laplace corresponding to the value 14 ((k+

1)2 − 4).


Corollary 3.13. The function

Fh (x, a) =cosh

(dh(x,a)(k+1)

2

)|x− Pa− a2 coshdh (x, a) e2| =

cosh(

dh(x,a)(k+1)2

)a2 sinh dh (x, a)

satisfies the equation∂2f

∂r2h+ 2 coth rh

∂f

∂rh+ γf = 0

with γ = 14 (4 − (k + 1)2) outside x = a and x

k−12

2 Fh (x, a) is k-hyperbolic outsidex = a.

Using (3.4), we may directly compute the corresponding k-hypermonogenicfunction.

Theorem 3.14. Denote rh = dh (x, a) and set s = k+12 . The function

hk (x, a) = as+12 xs−1

2 wk (x, a) p (x, a)

is paravector valued k-hypermonogenic with respect to x outside x = a when

wk (x, a) = (s− 1) cosh (srh) e2x− Pa

a2+ s cosh ((s− 1) rh)

and the function

p (x, a) =(x− c (x, a))

−1

x2 |x− c (x, a)|is hypermonogenic with respect to x.

Proof. Denote s = k+12 . Applying the previous corollary and Proposition 3.4, we

note that the function

g (x2, rh) = −as−12 xs−1

2 cosh (srh)

sinh rh

is k-hyperbolic harmonic and hk = Dxg (x2,rh) is k-hypermonogenic.We just make

simple calculations

hk (x, a)

as−12 xs−1

2

= (s− 1)e2 cosh (srh)

x2 sinh rh−(s sinh (srh)− cosh (srh) coth rh

sinh rh

)D

xrh.

Applying Lemma 3.5, we obtain

x2Dxrh

a22 sinh2 rh

=x− c (x, a)

|x− c (x, a)|3 =(x− c (x, a))−1

|x− c (x, a)|and

x− c (x, a)

a2

(x− c (x, a))−1

|x− c (x, a)| =1

a2 |x− c (x, a)|=

1

a22 sinh rh.


Hence we obtain

hk (x, a)

as+12 xs−1

2

= wk (x, a)(x− c (x, a))−1

x2 |x− c (x, a)|and

wk (x, a) = (s− 1) cosh (srh) e2x− c (x, a)

a2− s sinh rh sinh (srh) + cosh rh cosh (srh)

= (s− 1) cosh (srh) e2x− Pa

a2+ s (cosh rh cosh (srh)− sinh rh sinh (srh)) .

Using the rule of hyperbolic cosines, we conclude


a2+ s cosh ((s− 1) rh)

and

hk (x, a) = as+12 xs−1

2 wk (x, a) p (x, a) .

Applying [11], we infer that the function p (x, a) is hypermonogenic, completingthe proof. �

Denote the real surface measure by dS and a real weighted volume measure by

dmk =1

xk2

dm.

The generalized Stokes theorem for Mk operators is the following result.

Theorem 3.15 ([2]). Let Ω be an open subset of R3+ (or R3

−) and K ⊂ Ω be a

smoothly bounded compact set with the outer unit normal field ν. If f, g ∈ C1 (Ω,H),then ∫

∂K

gνfdS

xk2

=

∫K

((M r

kg) f + gM lkf −

k

x2P (gf ′) en

)dm

xk2

.

By taking the P -part from both sides of the equation of the previous theoremwe obtain.

Theorem 3.16 ([2]). Let Ω be an open subset of R3\ {x2 = 0} and K ⊂ Ω be asmoothly bounded compact set with the outer unit normal field ν. If f, g ∈ C1 (Ω,H),then ∫

∂K

P (gνf)dS

xk2

=

∫K

P((M r

kg) f + gM lkf) dmxk2

.

Theorem 3.17. Let Ω be an open subset of R3+ (or R3−) and K ⊂ Ω be a smoothly

bounded compact set with the outer unit normal field ν. Let s = k+12 and

hk (x, a) = as+12 xs−1

2 wk (x, a) p (x, a)


be the same function as in Theorem 3.14. If f is k-hypermonogenic in Ω anda ∈ K, then

Pf (a) =1

4π

∫∂K

P (hk (x, a) νf (x))dS

xk2

.

Proof. Using Theorem 3.16 we obtain∫∂K

P (hk (x, a) νf)dS

xk2

=

∫∂(K\Bh

r (a))

P (hk (x, a) νf)dS

xk2

−∫∂Brh

(a)

P (hk (x, a) νf)dS

xk2

= −∫∂Brh

(a)

P (hk (x, a) νf)dS

xk2

= −P(∫

∂Brh(a)

hk (x, a) νfdS

xk2

).

Since the hyperbolic ball ∂Brh (a) is the Euclidean ball with the Euclidean centerca (rh) = Pa+ a2 cosh rhe2 and the radius r = an sinh rh we have

ν (x) =x− ca (rh)

a2 sinh rh

and we deduce

P

(∫∂Brh

(a)

hk (x, a) νfdS

xk2

)

=1

a22 sinh2 rh

P

(∫∂Brh

(a)

ak+32

2

((s− 1) cosh (srh) e2

x− Pa

a2

+s cosh ((s− 1) rh)

)fdS

xk+32

2

).

If rh → 0 then

limrh→0

P

(∫∂Brh

(a)

hk (x, a) νfdS

xk2

)= 4πPf (a) . �

The Q-part satisfies the following generalized Stokes theorem.


−) and K ⊂ Ω be asmoothly bounded compact set with the outer unit normal field ν. If f, g ∈ C1 (Ω,H),then ∫

∂K

gνfdS =

∫K

((M r

−kg)f + gM l

kf +k

xnQ (gf ′)

)dm.

Applying the operator Q to the previous result, we directly conclude thefollowing result:



−) and K ⊂ Ω be a

smoothly bounded compact set with the outer unit normal field ν. If f, g ∈ C1 (Ω,H),then ∫

∂K

Q (gνf) dS =

∫K

Q((M r

−kg)f + gM l

kf)dm.

Theorem 3.20. Let Ω be an open subset of R3+ (or R3−) and K ⊂ Ω be a smoothly

bounded compact set with the outer unit normal field ν. Let s = k+12 . The function

v−k (x, a) = as+12 x−s

2 w−k (x, a) p (x, a)

is paravector valued −k-hypermonogenic outside x = a with respect to x, when

w−k (x, a) = −s cosh ((s− 1) rh) e2x− Pa

a2− (s− 1) cosh (srh) .

If f is k-hypermonogenic in Ω and a ∈ K, then

Qf (a) =1

4π

∫∂K

Q (v−k (x, a) νf (x)) dS.

The shifted Euclidean kernel may also be computed using the hat involution.

Lemma 3.21 ([5]). If x and a belong to R3+ then

p (x, a) = 4x2(x− a)

−1

|x− a| e2(x− a)

−1

|x− a| = 4x2(x− a)

−1

|x− a| e2(x− a)

−1

|x− a| .

Theorem 3.22. Let Ω be an open subset of R3+\ {x2 = 0} and K ⊂ Ω be a smoothly

bounded compact set with the outer unit normal field ν. If f is k-hypermonogenicin Ω and y ∈ K, then

f (a) =a22π

∫∂K

gk (x, a)((x− a)

−1ν (x) f (x)− (x− a)

−1ν (x) f (x)

)dS

|x− a| |x− a|where s = k+1

2 and

gk (x, a) = as−12 x1−s

2 ((s− 1) cosh (srh)− s cosh ((s− 1) rh)) .

Proof. Applying Theorems 3.17 and 3.22 we deduce

8πf (a) =

∫∂K

2P (hk (x, a) νf (x))dS

xk2

+

∫∂K

2Q (v−k (x, a) νf (x)) dSe2

=

(∫∂K

hk (x, a) νf (x)dS

xk2

+

∫∂K

hk (x, a)ν (x) f (x)dS

xk2

)+

(∫∂K

v−k (x, a) νf (x) dS −∫∂K

v−k (x, a)ν (x) f (x) dS

)=

∫∂K

ak+32

2 (wk (x, a) + w−k (x, a)) p (x, a) νf (x)dS

xk+12

2

+

∫∂K

ak+32

2

(wk (x, a)− w−k (x, a)

)p (x, a)ν (x) f (x)

dS

xk+12

2


where


a2+ s cosh ((s− 1) rh) ,

w−k (x, a) = −s cosh ((s− 1) rh) e2x− Pa

a2− (s− 1) cosh (srh) .

Since

e2x− Pa

a2− 1 = e2

x− Pa+ a2e2a2

= e2x− a

a2we have

wk (x, a) + w−k (x, a) = ((s− 1) cosh (srh)− s cosh ((s− 1) rh)) e2x− a

a2.

Applying

e2x− Pa

a2+ 1 = e2

x− Pa− a2e2a2

= e2x− a

a2we conclude that

wk (x, a)− w−k (x, a) = (s− 1) cosh (srh)− s cosh ((s− 1) rh)e2 (x− a)

a2,

completing the proof. �

Theorem 3.23. Let Ω be an open subset of R3+ and K ⊂ Ω a smoothly bounded

compact set with outer unit normal field ν. If f is k-hypermonogenic in Ω anda ∈ intK then

f (a) =1

4π

∫∂K

gk (x, a) p (a, x) (Q (xν(x)f ′ (x)) + aQ (ν(x)f ′(x))) dSx

where the function

p (a, x) =(a− c (a, x))−1

a2 |a− c (a, x)|is hypermonogenic with respect to a and

gk (x, a) = as−12 x1−s

2 ((s− 1) cosh (srh)− s cosh ((s− 1) rh))

for s = k+12 .

Proof. Decomposing

ν (x) f (x) = P (ν (x) f (x)) +Q (ν (x) f (x)) e2

and

ν (x)f (x) = ν (x) f (x) = P (ν (x) f (x))−Q (ν (x) f (x)) e2

we obtain

2πf (a) = a2

∫∂K

gk (x, a)(x− a)−1 − (x− a)−1

|a− x| |a− x| P (ν (x) f (x)) dσx

+ a2

∫∂K

gk (x, a)(x− a)

−1+ (x− a)

−1

|a− x| |a− x| e2Q′ (ν (x) f (x)) dσx.


If we factor (x− a)−1 to the left side of the difference (x− a)−1 − (x− a)−1 weobtain

(x− a)−1 − (x− a)−1 = (x− a)−1(1− (x− a) (x− a)−1

).

Then factoring (x− a)−1

to the right side of the difference(1− (x− a) (x− a)

−1)

we infer further

(x− a)−1 − (x− a)

−1= (x− a)

−1(1− (x− a) (x− a)

−1)

= (x− a)−1 (x− a− x+ a) (x− a)−1

= (x− a)−1

(x− x) (x− a)−1

= 2x2 (x− a)−1 e2 (x− a)−1 .

Applying Lemma 3.21, we obtain

a2(x− a)

−1 − (x− a)−1

|a− x| |a− x| =2x2a2 (a− x)

−1e2 (a− x)

−1

|a− x| |a− x| =x2 (a− c (a, x))

−1

2a2 |a− c (a, x)| .

Decomposing x = Px+ x2e2 and x = Px− x2e2 we obtain

(x− a)−1

(Px+ x2e2 − a) = 1 = (x− a)−1

(Px− x2e2 − a).

Hence

(x− a)−1 x2e2 − (x− a)−1 (a− Px) = − (x− a)−1 a2e2 − (x− a)−1 (a− Px) ,

which implies((x− a)

−1+ (x− a)

−1)x2e2 =

((x− a)

−1 − (x− a)−1)(a− Px)

and further((x− a)

−1+ (x− a)

−1)x2a2e2

|a− x| |a− x| = a2(x− a)

−1 − (x− a)−1

|a− x| |a− x| (a− Px)

=x2 (a− c (a, y))−1

2a2 |a− c (a, y)| (a− Px) .

Combining the previous equalities we have

2πf (a) =

∫∂K

gk (x, a) (a− c (a, x))−1

2a2 |a− c (a, x)| x2P (ν (x) f (x)) dS

+

∫∂K

gk (x, a)(a− c (a, x))

−1

2a2 |a− c (a, x)| (a− Px)Q′ (ν (xy) f (x)) dS

=

∫∂K

gk (x, a)(a− c (a, x))−1

2a2 |a− c (a, x)|u (x) dS

+

∫∂K

gk (x, a)(a− c (a, x))

−1

2a2 |a− c (a, x)|aQ′ (ν (x) f (x)) dS,


where

u (x) = x2P (ν (x) f (x))− PxQ′ (ν (x) f (x)) .

Applying (2.3) and (2.4) we deduce

−PxQ′ (ν (x) f (x)) + x2P (ν (x) f (x)) = PxQ (ν′ (x) f ′ (x)) +QxP ′ (ν′ (x) f ′ (x))

= Q (xν′ (x) f ′ (x)) ,

completing the proof. �

Conclusion. We have proved integral formulas for k-hypermonogenic functionsin R3. In consecutive papers we are going to study properties of correspondingintegral operators and generalize results to higher dimensions.

Acknowledgment

N. Vieira was supported by Portuguese funds through the CIDMA – Center forResearch and Development in Mathematics and Applications, and the PortugueseFoundation for Science and Technology (“FCT – Fundacao para a Ciencia e aTecnologia”), within project PEst-OE/MAT/UI4106/2014.

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Sirkka-Liisa Eriksson and Heikki OrelmaDepartment of MathematicsTampere University of TechnologyP.O.Box 553FI-33101 Tampere, Finlande-mail: [email protected]

[email protected]

Nelson VieiraDepartment of MathematicsUniversity of Aveiro Universitario de Santiago3810-193 Aveiro, Portugale-mail: [email protected]






Spectral Properties of Compact NormalQuaternionic Operators

Riccardo Ghiloni, Valter Moretti and Alessandro Perotti

Abstract. General, especially spectral, features of compact normal operatorsin quaternionic Hilbert spaces are studied and some results are establishedwhich generalize well-known properties of compact normal operators in com-plex Hilbert spaces. More precisely, it is proved that the norm of such anoperator always coincides with the maximum of the set of absolute values ofthe eigenvalues (exploiting the notion of spherical eigenvalue). Moreover thestructure of the spectral decomposition of a generic compact normal operatorT is discussed also proving a spectral characterization theorem for compactnormal operators.

Mathematics Subject Classification (2010).46S10, 47C15, 47B07, 30G35, 81R15.

Keywords. Compact operators, quaternionic Hilbert spaces.

1. Introduction

Theory of linear operators in quaternionic Hilbert spaces is a well-establishedtopic of functional analysis with many applications in physics, especially quan-tum mechanics (see the introduction of [6] for a wide discussion). As in complexfunctional analysis, compact operators play a relevant role as they share featuresboth with generic operators in infinite-dimensional spaces and with matrices infinite-dimensional spaces. This intermediate role is particularly evident regardingspectral analysis of normal compact operators. In fact, these operators in infinite-dimensional (complex or quaternionic) Hilbert spaces, on the one hand admit apure point spectrum (except, perhaps, for 0), on the other hand their spectralexpansion needs a proper infinite Hilbertian basis. This paper is devoted to focuson these peculiar properties exploiting the general framework established in [6].

Work partially supported by GNSAGA and GNFM of INdAM, MIUR-PRIN project “Varieta

reali e complesse: geometria, topologia e analisi armonica” and MIUR-FIRB project “GeometriaDifferenziale e Teoria Geometrica delle Funzioni”.

Switzerland

134 R. Ghiloni, V. Moretti and A. Perotti

The notion of spectrum of an operator on quaternionic Hilbert spaces hasbeen introduced only few years ago [2] in the more general context of quaternionicBanach modules. It has been a starting point for developing functional calculus forthe classes of slice and slice regular functions on a quaternionic space (see [2, 6]).

Let H be a (right) quaternionic Hilbert space (we refer to Section 2 for basicdefinitions and to [6] for more details), let B(H) be the set of right linear op-erators on H and let B0(H) be the set of right linear compact operators on H.In [5] some properties of compact operators on quaternionic Hilbert spaces werestudied. In particular, it was shown that the spherical spectrum (cf. Section 2.3for complete definitions) of a compact operator T contains only the eigenvalues ofT and possibly 0, and the set of eigenvectors relative to a non-zero eigenvalue q isfinite dimensional. Another result, similar to what occurs for compact operatorsin complex Hilbert spaces, is conjectured in [5]:

Conjecture. If T ∈ B0(H) is self-adjoint, then either ‖T ‖ or −‖T ‖ is an eigenvalueof T .

In the following we will prove the conjecture for the more general class of normalcompact operators on a quaternionic Hilbert space. We also prove the spectral de-composition theorem for normal compact operators and its converse. The complexHilbert space versions of these results can be found for example in [8, §3.3].1.1. Main theorems

As recalled in Section 2.3, the set of eigenvalues of a linear operator T coincideswith the spherical point spectrum, denoted by σpS(T ). We can then rephrase theconjecture in the following way.

Theorem 1.1. Given any normal operator T ∈ B0(H) with spherical point spectrumσpS(T ), there exists λ ∈ σpS(T ) such that:

|λ| = max{|μ| | μ ∈ σS(T )} = ‖T ‖ . (1.1)

The next result is the spectral decomposition for normal compact operators.In the finite-dimensional case, where the compactness requirement is empty, theresult is well known [7] (see [4] for an ample exposition and further references).

Theorem 1.2. Given a normal operator T ∈ B0(H) with spherical point spectrumσpS(T ), there exists a Hilbert basis N ⊂ H made of eigenvectors of T such that:

Tx =∑z∈N

zλz〈z|x〉 for each x ∈ H, (1.2)

where λz ∈ H is an eigenvalue relative to the eigenvector z and, if λz = 0 onlya finite number of distinct other elements z′ ∈ N verify λz = λz′ , moreover thevalues λz are at most countably many.

The set Λ of eigenvalues λz with z ∈ N has the property that for every ε > 0there is a finite set Λε ⊂ Λ with |λ| < ε if λ ∈ Λε (following [8] we say that theeigenvalues “vanish at infinity”). Thus 0 is the only possible accumulation pointof Λ. If H is infinite dimensional, then 0 belongs to σS(T ).

Spectral Properties of Compact Normal Quaternionic Operators 135

Remark 1.3. Let S denote the two-dimensional sphere of imaginary units in H:

S := {q ∈ H | q2 = −1}.For ı ∈ S, let Cı be the real subalgebra of H generated by ı. We will see in Section2 that to every normal operator T can be associated an anti self-adjoint operatorJ . Along with a chosen imaginary unit ı ∈ S, J defines (cf. Definition 2.5) twoCı-Hilbert subspaces of H, denoted by HJı

+ and HJı− .

In the preceding theorem, for every imaginary unit ı, it is possible to chooseN such that:

{λz | z ∈ N} \ {0} = (σpS(T ) \ {0}) ∩ Cı (1.3)

as well as N ⊂ HJı+ . With a conjugation, in general losing the condition N ⊂ HJı

+ ,

one can always have λz ∈ C+ı but N ⊂ HJı

+ ∪ HJı− .

The spectral theorem for compact operators has the following converse. Tostate it, we need to recall a definition. Given a subset K of C, we define thecircularization ΩK of K (in H ) by setting

ΩK := {α+ jβ ∈ H |α, β ∈ R, α+ iβ ∈ K, j ∈ S}. (1.4)

Theorem 1.4. Let T ∈ B(H). Assume that there exist a Hilbert basis N of H anda map N � z �→ λz ∈ H satisfying the following requirements:

(i) Tx =∑

z∈N zλz〈z|x〉 for every x ∈ H.(ii) For every z ∈ N such that λz = 0, only a finite number of distinct other

elements z′ ∈ N verify λz = λz′ ;(iii) The set Λ is countable at most;(iv) For every ε > 0, there is a finite set Λε ⊂ Λ with |λ| < ε if λ ∈ Λε.

Under these conditions T is normal and compact and

σS(T ) \ {0} = ΩΛ \ {0} .Remark 1.5. The structure of the whole spherical spectrum (see Definition 2.7) ofa compact operator T ∈ B0(H) has been studied in [5, Corollary 2]:

σS(T ) \ {0} = σpS(T ) \ {0} .If T is normal, then its spherical residual spectrum (cf. Section 2.3 for definitions)is empty. Therefore in this case if 0 ∈ σS(T )\σpS(T ) then 0 belongs to the sphericalcontinuous spectrum σcS(T ).

2. Quaternionic Hilbert spaces

We recall some basic notions about quaternionic Hilbert spaces (see, e.g., [1]). LetH denote the skew field of quaternions. Let H be a right H-module. H is calleda quaternionic pre-Hilbert space if there exists a Hermitian quaternionic scalarproduct H× H � (u, v) �→ 〈u|v〉 ∈ H satisfying the following three properties:

• Right linearity: 〈u|vp+ wq〉 = 〈u|v〉p+ 〈u|w〉q if p, q ∈ H and u, v, w ∈ H.

• Quaternionic Hermiticity: 〈u|v〉 = 〈v|u〉 if u, v ∈ H.• Positivity: If u ∈ H, then 〈u|u〉 ∈ R+ and u = 0 if 〈u|u〉 = 0.


We can define the quaternionic norm by setting

‖u‖ :=√〈u|u〉 ∈ R+ if u ∈ H.

Definition 2.1. A quaternionic pre-Hilbert space H is said to be a quaternionicHilbert space if it is complete with respect to its natural distance d(u, v) := ‖u−v‖.Example 2.2. The space Hn with scalar product 〈u, v〉 =

∑ni=1 uivi is a finite-

dimensional quaternionic Hilbert space.

Let H be a quaternionic Hilbert space.

Definition 2.3. A right H-linear operator is a map T : D(T ) −→ H such that:

T (ua+ vb) = (Tu)a+ (Tv)b if u, v ∈ D(T ) and a, b ∈ H,

where the domain D(T ) of T is a (not necessarily closed) right H-linear subspaceof H.

It can be shown that an operator T : D(T ) −→ H is continuous if and onlyif it is bounded, i.e., there exists K ≥ 0 such that

‖Tu‖ ≤ K‖u‖ for each u ∈ D(T ).

Let ‖T ‖ := supu∈D(T )\{0}‖Tu‖‖u‖ = inf{K ∈ R | ‖Tu‖ ≤ K‖u‖ ∀u ∈ D(T )}. The set

B(H) of all bounded operators T : H −→ H is a complete metric space w.r.t. themetric D(T, S) := ‖T − S‖,

Many assertions that are valid in the complex Hilbert spaces case, continue tohold for quaternionic operators. We mention the uniform boundedness principle,the open map theorem, the closed graph theorem, the Riesz representation theoremand the polar decomposition of operators.

As in the complex case, a linear operator T : H → H is called compactif it maps bounded sequences to sequences that admit convergent subsequences.We refer to [5] for some properties of compact operators on quaternionic Hilbertspaces. In particular, B0(H) is a closed bilateral ideal of B(H) and is closed underadjunction ([5, Theorem 2]).

2.1. Left scalar multiplications

It is possible to equip a (right) quaternionic Hilbert space H with a left multi-plication by quaternions. It is a non-canonical operation relying upon a choice ofa preferred Hilbert basis. So, pick out a Hilbert basis N of H and define the leftscalar multiplication of H induced by N as the map H × H � (q, u) �→ qu ∈ Hgiven by

qu :=∑

z∈N zq〈z|u〉 if u ∈ H and q ∈ H.

For every q ∈ H, the map Lq : u �→ qu belongs to B(H). Moreover, the mapLN : H −→ B(H), defined by setting LN (q) := Lq is a norm-preserving realalgebra homomorphism.


The set B(H) is always a real Banach C∗-algebra with unity. It suffices toconsider the right scalar multiplication (Tr)(u) = T (u)r for real r and the adjun-ction T �→ T ∗ as ∗-involution. By means of a left scalar multiplication, it can begiven the richer structure of quaternionic Banach C∗-algebra.

Theorem 2.4 ([6]§3.2). Let H be a quaternionic Hilbert space equipped with a leftscalar multiplication. Then the set B(H), equipped with the pointwise sum, withthe scalar multiplications defined by

(qT )u := q(Tu) and (Tq)(u) := T (qu),

with the composition as product and with T �→ T ∗ as ∗-involution, is a quaternionictwo-sided Banach C∗-algebra with unity.

Observe that the map LN gives a ∗-representation of H in B(H).

2.2. Imaginary units and complex subspaces

Consider a quaternionic Hilbert space H equipped with a left scalar multiplicationH � q �→ Lq. For short, we write Lqu = qu. For every imaginary unit ı ∈ S, theoperator J := Lı is anti self-adjoint and unitary; that is, it holds:

J∗ = −J and J∗J = I.

It holds also the converse statement: if an operator J ∈ B(H) is anti self-adjointand unitary, then J = L′

ı for some left scalar multiplication of H (see [6, Proposi-tion 3.8]).

In the following, we also need a definition known from the literature [3].

Definition 2.5. Let J ∈ B(H) be an anti self-adjoint and unitary operator and letı ∈ S. Let Cı denote the real subalgebra of H generated by ı; that is, Cı := {α+ıβ ∈H |α, β ∈ R}. Define the Cı-complex subspaces HJı

+ and HJı− of H associated with

J and ı by settingHJı

± := {u ∈ H | Ju = ±uı}.Remark 2.6. HJı

± are closed subsets of H, because u �→ Ju and u �→ ±uı arecontinuous. However, they are not (right H-linear) subspaces of H. Note also thatthe space H admits the direct sum decomposition

H = HJı+ ⊕ HJı

− ,

with projections H � x �→ P±(x) := 12 (x ∓ Jxı) ∈ HJı± .

2.3. Resolvent and spectrum

It is not clear how to extend the definitions of spectrum and resolvent in quater-nionic Hilbert spaces. Let us focus on the simpler case of eigenvalues of a boundedright H-linear operator T . Without fixing any left scalar multiplication of H, theequation determining the eigenvalues reads as follows:

Tu = uq.

Here a drawback arises: if q ∈ H \R is fixed, the map u �→ uq is not right H-linear.Consequently, the eigenspace of q cannot be a right H-linear subspace. Indeed, if


λ = 0, uλ is an eigenvector of λ−1qλ instead of q itself. As a second guess, onecould decide to deal with quaternionic Hilbert spaces equipped with a left scalarmultiplication and require that

Tu = qu.

Now both sides are right H-linear. However, this approach is not suitable forphysical applications, where self-adjoint operators should have real spectrum. Wecome back to the former approach and accept that each eigenvalue q brings a wholeconjugation class of the quaternions, the eigensphere

Sq := {λ−1qλ ∈ H |λ ∈ H \ {0}}.We adopt the viewpoint introduced in [2] for quaternionic two-sided Banach

modules. Given an operator T : D(T ) −→ H and q ∈ H, let

Δq(T ) := T 2 − T (q + q) + I|q|2.Definition 2.7. The spherical resolvent set of T is the set ρS(T ) of q ∈ H suchthat:

(a) Ker(Δq(T )) = {0}.(b) Range(Δq(T )) is dense in H.

(c) Δq(T )−1 : Range(Δq(T )) −→ D(T 2) is bounded.

The spherical spectrum σS(T ) of T is defined by σS(T ) := H \ ρS(T ). Itdecomposes into three disjoint circular (i.e., invariant by conjugation) subsets:

(i) the spherical point spectrum of T (the set of eigenvalues):

σpS (T ) := {q ∈ H |Ker(Δq(T )) = {0}}.(ii) the spherical residual spectrum of T :

σrS (T ) :={q ∈ H

∣∣∣Ker(Δq(T )) = {0}, Range(Δq(T )) = H}.

(iii) the spherical continuous spectrum of T :

σcS (T ) :={q ∈ H

∣∣Δq(T )−1 is densely defined but not bounded

}.

The spherical spectral radius of T is defined as

rS(T ) := sup{|q|∣∣ q ∈ σS(T )

} ∈ R+ ∪ {+∞}.

2.4. Spectral properties

The spherical resolvent and the spherical spectrum can be defined for boundedright H-linear operators on quaternionic two-sided Banach modules in a form sim-ilar to that introduced above (see [2]). Several spectral properties of boundedoperators on complex Banach or Hilbert spaces remain valid in that general con-text. Here we recall some of these properties in the quaternionic Hilbert setting(cf. Theorem 4.3 in [6]).


Theorem 2.8 ([6]§4.1). Let H be a quaternionic Hilbert space and let T ∈ B(H).Then

(a) rS(T ) ≤ ‖T ‖.(b) σS(T ) is a non-empty compact subset of H.(c) Let P ∈ R[X ]. Then, if T is self-adjoint, the following spectral map property

holds:σS(P (T )) = P (σS(T )).

(d) Gelfand’s spectral radius formula holds:

rS(T ) = limn→+∞ ‖T

n‖1/n.In particular, if T is normal (i.e., TT ∗ = T ∗T ), then rS(T ) = ‖T ‖.Regardless different definitions with respect to the complex Hilbert space

case, the notions of spherical spectrum and resolvent set enjoy some propertieswhich are quite similar to those for complex Hilbert spaces. Other features, con-versely, are proper to the quaternionic Hilbert space case. First of all, it turns outthat the spherical point spectrum coincides with the set of eigenvalues of T .

Proposition 2.9. Let H be a quaternionic Hilbert space and let T : D(T ) −→ H bean operator. Then σpS(T ) coincides with the set of all eigenvalues of T .

The subspace Ker(Δq(T )) has the role of an eigenspace of T . In particular,Ker(Δq(T )) = {0} if and only if Sq is an eigensphere of T .

Theorem 2.10. Let T be an operator with dense domain on a quaternionic Hilbertspace H.

(a) σS(T ) = σS(T∗).

(b) If T ∈ B(H) is normal, then(i) σpS (T ) = σpS (T

∗).(ii) σrS (T ) = σrS (T

∗) = ∅.(iii) σcS (T ) = σcS (T

∗).(c) If T is self-adjoint, then σS(T ) ⊂ R and σrS (T ) is empty.(d) If T is anti self-adjoint, then σS(T ) ⊂ Im(H) and σrS (T ) is empty.(e) If T ∈ B(H) is unitary, then σS(T ) ⊂ {q ∈ H | |q| = 1}.(f) If T ∈ B(H) is anti self-adjoint and unitary, then σS(T ) = σpS(T ) = S.

It can be shown that, differently from operators on complex Hilbert spaces,a normal operator T on a quaternionic space is unitarily equivalent to T ∗.

2.5. Compact operators

Compact operators have some peculiar spectral properties. Some of them wereinvestigated in [5]. In particular, if T ∈ B0(H) and q ∈ σpS (T )\{0} is an eigenvalue,then Ker(Δq(T )) has finite dimension [5, Theorem 3]. Moreover, the sphericalspectrum of T ∈ B0(H) consists only of the eigenvalues of T and (possibly) 0 (cf.[5, Corollary 2]):

σS(T ) \ {0} = σpS(T ) \ {0}.


2.6. Slice nature of normal operators

We recall the “slice” character of H:

• H =⋃

j∈SCj where Cj is the real subalgebra 〈j〉 ! C.

• Cj ∩ Cκ = R for every j, κ ∈ S with j = ±κ.This decomposition of H has an “operatorial” counterpart on a quaternionic

Hilbert space. It was established in Theorem 5.9 of [6].

Theorem 2.11 ([6]§5.4). Given any normal operator T ∈ B(H), there exist threeoperators A,B, J ∈ B(H) such that:

(i) T = A+ JB.(ii) A is self-adjoint and B is positive.(iii) J is anti self-adjoint and unitary.(iv) A, B and J commute mutually.

Furthermore, it holds:

• A and B are uniquely determined by T : A = (T + T ∗)12 and B = |T − T ∗| 12 .• J is uniquely determined by T on Ker(T − T ∗)⊥.

(where for S ∈ B(H), |S| denotes the operator defined as the square root of thepositive operator S∗S).

In the following, we denote by σ(B) and ρ(B) the standard spectrum andresolvent set of a bounded operator B of a complex Hilbert space, respectively.

Proposition 2.12 ([6] §5.4). Let H be a quaternionic Hilbert space, let T ∈ B(H)be a normal operator, let J ∈ B(H) be an anti self-adjoint and unitary operatorsatisfying TJ = JT , T ∗J = JT ∗, let ı ∈ S and let HJı± be the complex subspaces ofH associated with J and ı (see Definition 2.5). Then we have that

(a) T (HJı+ ) ⊂ HJı

+ and T ∗(HJı+ ) ⊂ HJı

+ .

Moreover, if T |HJı+

and T ∗|HJı+

denote the Cı-complex operators in B(HJı+ ) obtained

restricting respectively T and T ∗ to HJı+ , then it holds:

(b) (T |HJı+)∗ = T ∗|HJı

+.

(c) σ(T |HJı+) ∪ σ(T |HJı

+) = σS(T ) ∩ Cı. Here σ(T |HJı

+) is considered as a subset

of Cı via the natural identification of C with Cı induced by the real vectorisomorphism C � α+ iβ �→ α+ ıβ ∈ Cı.

(d) σS(T ) = ΩK, where K := σ(T |HJı+).

An analogous statement holds for HJı− .

3. Proofs of the main results

3.1. Proof of Theorem 1.1

If T = 0 there is nothing to prove, since 0 ∈ σpS(T ) and ‖T ‖ = 0 in that case. So,we henceforth assume that T = 0. Since T ∈ B(H) is normal, Theorem 2.11 assuresthe existence of an anti self-adjoint unitary right H-linear operator J : H→ H,


commuting with T and T ∗ and fulfilling T = (T + T ∗)12 + J |T − T ∗| 12 . Next,if HJı

+ is the complex subspace associated with an imaginary unit ı ∈ S as inDefinition 2.5, it turns out that T is the unique right H-linear operator, definedon H, whose restriction to HJı

+ coincides with the complex-linear operator S :=

T |HJı+

: HJı+ → HJı

+ (it immediately arises from Propositions 3.11 and 5.11 in [6]).

We also know that ‖S‖ = ‖T ‖, in view of Proposition 3.11 in [6].By hypotheses T is compact and thus S is compact as well as we go to prove.

If {un}n∈N ⊂ HJı+ is a bounded sequence of vectors, it is a bounded sequence of vec-

tors of H too, and thus the sequence {Tun}n∈N admits a subsequence {Tunk}k∈N

converging to some v ∈ H, because T is compact. However, since HJı+ is closed

(because of its definition and the fact that J is continuous), we also have thatv ∈ HJı

+ and that {Sunk}k∈N converges to v ∈ HJı

+ , because Tun = Sun. We have

found that, for every bounded sequence {un}n∈N ⊂ HJı+ , there is a subsequence of

{Sun}n∈N converging to some v ∈ HJı+ . Thus S is compact.

To go on, Lemma 3.3.7 in [8] entails that there exists λ ∈ σp(S) with |λ| =‖S‖. Notice that λ = 0 otherwise S = 0 and thus T = 0 by uniqueness of theextension of S. Finally, point (d) of Proposition 2.12 implies that λ ∈ σS(T ).Since T is compact, by Corollary 2 of [5], we have that λ ∈ σpS(T ). Summingup, we have obtained that there is λ ∈ σpS(T ) with |λ| = ‖S‖ = ‖T ‖, where theabsolute value is, indifferently, that in C or that in H. The remaining identity in(1.1) is now equivalent to: sup{|μ| | μ ∈ σS(T )} = ‖T ‖, i.e., rS(T ) = ‖T ‖. In thisform, it was proved in point (d) of Theorem 2.8.


Fix an imaginary unit ı ∈ S and consider the normal compact operator S : HJı+ →

HJı+ as in the proof of Theorem 1.1. As a consequence of Theorem 3.3.8 in [8],

there exists a Hilbert basis N ⊂ HJı+ made of eigenvectors of S and a map N �

z �→ λz ∈ Cı such that each λz is an eigenvalue of S in Cı relative to z ∈ N and,if λz = 0, only a finite number of distinct other elements z′ ∈ N verify λz = λz′ .Moreover, the values λz are at most countably many. We know by Lemma 3.10(b)in [6] that N is also a Hilbert basis of H, so that, if x ∈ H, then

x =∑z∈N

z〈z|x〉 .

Since T is continuous and Tz = Sz = zλz, we have:

Tx =∑z∈N

zλz〈z|x〉 .

From Theorem 3.3.8 in [8], we also get that the set {λz}z∈N of the eigenvaluesof S, and therefore those of T , vanish at infinity. Thus 0 is the only possibleaccumulation point of Λ. If H is not finite dimensional and 0 is not an eigenvalueof T , then 0 must be an accumulation point of Λ, since every set of eigenvectorsKer(Δλz (T )) has finite dimension if λz = 0. Since σS(T ) is closed, in any case0 ∈ σS(T ).


Remark 3.1. Since λz ∈ Cı for each z ∈ N , equation (1.3) follows from Corollary 2in [5].


Fix an imaginary unit ı ∈ S. For each eigenvector z ∈ N , with non real eigenvalueλz ∈ H, we can choose a unit quaternion μz such that μ−1

z λzμz belongs to theintersection of the eigensphere of λz with the complex plane Cı. This means thatz′ := zμz is still an eigenvector of T , with eigenvalue λz′ := μ−1

z λzμz ∈ Cı. Ifλz ∈ R, we set μz = 1. The set N ′ := {zμz | z ∈ N} is still a Hilbert basis of H,such that Tx =

∑z∈N ′ zλz〈z|x〉 for every x ∈ H.

The linear operator J defined by setting

Jx :=∑z∈N ′

zı〈z|x〉

is an anti self-adjoint and unitary operator on H (cf. Proposition 3.1 in [6]). Since

TJx =∑z∈N ′

zλz〈z|Jx〉 =∑z∈N ′

zλzı〈z|x〉

and

JTx =∑z∈N ′

zı〈z|Tx〉 =∑z∈N ′

zıλz〈z|x〉,

we have that J and T commute. Moreover, since T ∗x =∑

z∈N ′ zλz〈z|x〉, thesame holds for J and T ∗. Let HJı

± be the complex subspaces of H associated with

J and the imaginary unit ı ∈ S as in Definition 2.5. Observe that N ′ ⊂ HJı+ ,

since Jz = zı for each z ∈ N ′. Let j ∈ S be an imaginary unit orthogonal to ı.Then N ′j := {zj | z ∈ N ′} is a Hilbert basis for HJı− (cf. Lemma 3.10 in [6]). The

Cı-complex subspaces HJı± of H are invariant for T , since

JTu = TJu = TJ(∓Juı) = ±(Tu)ı for each u ∈ HJı± .

Let S± := T |HJı±

be the restrictions of T . Then S± are diagonalizable, since

S+x+ =∑z∈N ′

zλz〈z|x+〉 and S−x− =∑z∈N ′

zjλz〈zj|x−〉

for every x± ∈ HJı± . We can then apply Theorem 3.3.8 of [8] and obtain that S±are normal and compact. Using Proposition 3.11 of [6], we get the normality of T .

It remains to prove that T is compact. Recall from Remark 2.6 that H = HJı+ ⊕

HJı− , with projections P± defined by P±(x) = 1

2 (x∓Jxı) ∈ HJı± . If {xn}n∈N ⊂ H is a

bounded sequence of vectors, then also {P+xn}n∈N ⊂ HJı+ and {P−xn}n∈N ⊂ HJı

−are bounded sequences. Since S+ is compact, the sequence {S+P+xn}n∈N ad-mits a subsequence {S+P+xnk

}k∈N converging to some v+ ∈ HJı+ . Similarly, since

S− is compact, we can extract from the sequence {S−P−xnk}k∈N a subsequence

{S−P−xnkl}l∈N converging to some v− ∈ HJı− . Then the sequence {Txnkl

}l∈N con-verges to v := v+ + v−, since Tx = S+P+x+ S−P−x for each x ∈ H.


We have shown that for every bounded sequence {xn}n∈N ⊂ H there is asubsequence of {Txn}n∈N converging to some v ∈ H. Thus T is compact.

The last statement concerning the spectrum of T follows from Proposi-tion 2.12.

References

[1] Brackx, F., Delanghe, R., Sommen, F.: Clifford analysis, Research Notes in Mathe-matics, vol. 76. Pitman (Advanced Publishing Program), Boston, MA (1982).

[2] Colombo, F., Sabadini, I., Struppa, D.C.: Noncommutative functional calculus,Progress in Mathematics, vol. 289. Birkhauser/Springer Basel AG, Basel (2011). DOI10.1007/978-3-0348-0110-2. URL http://dx.doi.org/10.1007/978-3-0348-0110-2. The-ory and applications of slice hyperholomorphic functions.

[3] Emch, G.: Mecanique quantique quaternionienne et relativite restreinte. I. Helv. Phys.Acta 36, 739–769 (1963).

[4] Farenick, D.R., Pidkowich, B.A.F.: The spectral theorem in quaternions. Lin-ear Algebra Appl. 371, 75–102 (2003). DOI 10.1016/S0024-3795(03)00420-8. URLhttp://dx.doi.org/10.1016/S0024-3795(03)00420-8.

[5] Fashandi, M.: Compact operators on quaternionic Hilbert spaces. Facta Univ. Ser.Math. Inform. 28(3), 249–256 (2013).

[6] Ghiloni, R., Moretti, V., Perotti, A.: Continuous slice functional calculus in quater-nionic Hilbert spaces. Rev. Math. Phys. 25(4), 1350,006, 83 (2013).DOI 10.1142/S0129055X13500062.URL http://dx.doi.org/10.1142/S0129055X13500062.

[7] Jacobson, N.: Normal Semi-Linear Transformations. Amer. J. Math. 61(1), 45–58(1939). DOI 10.2307/2371384. URL http://dx.doi.org/10.2307/2371384.

[8] Pedersen, G.K.: Analysis now, Graduate Texts in Mathematics, vol. 118. Springer-Verlag, New York (1989). DOI 10.1007/978-1-4612-1007-8.URL http://dx.doi.org/10.1007/978-1-4612-1007-8.

Riccardo Ghiloni, Valter Moretti and Alessandro PerottiDepartment of MathematicsUniversity of TrentoI-38123, Povo-Trento, Italye-mail: [email protected]

[email protected]

[email protected]

http://dx.doi.org/10.1007/978-3-0348-0110-2

http://dx.doi.org/10.1016/S0024-3795(03)00420-8

http://dx.doi.org/10.1142/S0129055X13500062

http://dx.doi.org/10.2307/2371384

http://dx.doi.org/10.1007/978-1-4612-1007-8






Three-dimensional Quaternionic Analogueof the Kolosov–Muskhelishvili Formulae

Yuri Grigor’ev

Abstract. The aim of this work is to construct a three-dimensional quater-nionic generalization of the Kolosov–Muskhelishvili formulae. The theory ofMoisil–Theodoresco system in terms of regular quaternionic functions of re-duced quaternion variable is used. As applications the main problems of anelastic sphere equilibrium are solved.

Mathematics Subject Classification (2010). Primary 30G35; Secondary 74B05.

Keywords. Classical linear elasticity, functions of hypercomplex variables andgeneralized variables, regular quaternionic functions, Moisil–Theodoresco sys-tem, representation formulae.

1. Introduction

In two-dimensional problems of the theory of elasticity the methods of complexvariable theory are effectively used. In plane problems the basis of this is the repre-sentation of the general solution of the equilibrium equations in terms of two arbi-trary analytic functions called the Kolosov–Muskhelishvili formulae [29]. In axiallysymmetric problems different classes of generalized analytic functions of complexvariable [1,39], p- and (p, q)-analytic functions [34] are used. As a generalization ofthe method of complex functions in multidimensional problems the methods of hy-percomplex functions are developed [9,11,22–25]. For three-dimensional problemsof mathematical physics such an apparatus is the Moisil–Theodoresco system the-ory, which is developed as the theory of regular quaternion functions of incompletequaternion variable [15,19,22]. We note that this theory is covered by Clifford anal-ysis. In [28] the first quaternion solution of the equilibrium equations of the theoryof elasticity is obtained, but without proof of representation generality. In [4, 5]a spatial quaternionic analog of the complex Kolosov–Muskhelishvili formulae in

The author is partially sponsored by RFBR, grant N 12-01-00507-a.

Switzerland

146 Yu. Grigor’ev

base on the Papkovich–Neuber representation and polynomial solutions as applica-tions are obtained. In [6,7] a very special way for a hypercomplex function theoryand its application are presented. In [14] three-dimensional representation formu-lae were derived for every term of the Fourier series expansion of displacementcomponents in cylindrical coordinates. Some other attempts were made in [32,33].In [16–20, 31] some preliminary results of using a quaternion function method inthe theory of elasticity with solutions of some problems for an elastic sphere arepresented. In [26] the method of a regularization of integral equations of elasticityusing a Cauchy-type integral for the Moisil–Theodoresco system was proposed.In [27] some connections between equations of continua and hypercomplex func-tions are noticed. The connection between numerical collocation and quaternionicrepresentations of solutions in three-dimensional elasticity was found out in [36].Other representations of the 3D elasticity and thermo-elasticity equations throughregular quaternion functions were given in [37, 38].

In this paper a variant of three-dimensional quaternion generalization of theKolosov–Muskhelishvili formulae is proposed, which is effectively applied to solvethe basic problems of the theory of elasticity for the ball. It is shown that inparticular cases of plane and axially symmetric deformations this representationgoes into the Kolosov–Muskhelishvili and Solov’ev formulae.

2. Preliminaries and notations

Let i, j, k be the basic quaternions obeying the following rules of multiplication:

i2 = j2 = k2 = −1, ij = −ji = k, jk = −kj = i, ki = −ik = j.

An element q of the quaternion algebra H we write in the form q = q0+ iqx+ jqy+kqz = q0 +q, where q0, qx, qy, qz are the real numbers, q0 is called the scalar partof the quaternion, q = iqx + jqy + kqz is called the vector part of the quaternionq. The quaternion conjugation is denoted as q = q0 − q.

Let x, y, z be the Cartesian coordinates in the Euclidean space R3. Let Ω be adomain of R3 with a piecewise smooth boundary. A quaternion-valued function or,briefly, H-valued function f of a reduced quaternion variable r = ix+ jy+kz ∈ R3

is a mapping

f : Ω −→ H,

such that

f(r) = f0(r) + f(r) = f0(x, y, z) + ifx(x, y, z) + jfy(x, y, z) + kfz(x, y, z).

The functions f0, fx, fy, fz are real valued defined in Ω. Continuity, differen-tiability or integrability of f are defined coordinate-wisely. For continuously real-differentiable functions f : Ω ⊂ R3 −→ H, which we will denote for simplicityby f ∈ C1(Ω,H), the operator ∇ = i∂x + j∂y + k∂z is called the generalizedCauchy–Riemann operator.

Three-dimensional Quaternionic Analogue 147

According to R. Fueter [22] a function f is called left-regular in Ω if

∇f = 0, r ∈ Ω. (2.1)

A similar definition can be given for right-regular functions. From now on we useonly left-regular functions that, for simplicity, we call regular. With the vectorialnotations the regularity condition is given as follows:

∇f(r) = −∇ · f(r) +∇f0(r) +∇× f(r) = 0, (2.2)

where ∇f0, ∇ · f , ∇× f are the usual gradient, divergence and curl, respectively.Thus, the coordinate-wise representations of the regularity condition are given asfollows: ⎧⎪⎪⎪⎨⎪⎪⎪⎩

fx,x + fy,y + fz,z = 0,

f0,x + fz,y − fy,z = 0,

f0,y + fx,z − fz,x = 0,

f0,z + fy,x − fx,y = 0.

(2.3)

The system (2.3) is called the Moisil–Theodoresco system (MTS) [15, 30] andis a spatial generalization of the Cauchy–Riemann system (CRS). If we assumethat f depends only on two variables, for example, x and y, then the MTS (2.3)splits into two CRS and the complex functions f(ζ) = fx(x, y)− ify(x, y), g(ζ) =f0(x, y) − ifz(x, y) will be the analytic functions of complex variable ζ = x + iy.If the MTS is written in the cylindrical coordinates ρ, ϕ, z, then in the caseof axial symmetry the MTS splits into two generalized by Vekua [39] CRS andf(ζ) = f0(z, ρ)−ifϕ(z, ρ), g(ζ) = fz(z, ρ)−ifρ(z, ρ) will be the generalized analyticby Vekua functions of the complex variable ζ = z+ iρ. Exactly these functions areused in axially symmetric problems [1].

The equations of elastic equilibrium are called the Lame equations:

Lu ≡ (λ+ 2μ)∇(∇ · u)− μ∇× (∇× u) = 0. (2.4)

If we introduce the next notations

(λ+ 2μ)∇ · u = f0, −μ∇× u = f , (2.5)

then the Lame equation (2.4) is transformed into the MTS:

∇ · f = 0, ∇f0 +∇× f = 0, (2.6)

thus the quaternion function f = f0+f is regular. In the paper [28] it was indicatedthat such the connection between the Lame equation and quaternion functions wasfirst pointed by G. Moisil.

148 Yu. Grigor’ev

3. The radial integration operator

We will introduce the operator of radial integration Iα, acting according to therule:

Iαf = Iαf(r) =

1∫0

tαf(rt)dt =

1∫0

tαf(rt, θ, ϕ)dt,

here r, θ, ϕ are the spherical coordinates. With the help of the operator Iα it isconvenient to solve problems in star, relative to the origin of coordinates, regions,which we will denote as Ω∗. We will give some properties of the operator Iα, whichare used in this paper.

1. (r · ∇) Iα = Iα(r · ∇) = r∂rIα = Iαr∂r .

2. (r · ∇) Iαf = r∂rIαf = f − (α+ 1)Iαf, f ∈ C0

α;3.

IαIβf = IβIαf =

⎧⎨⎩(α− β)−1(Iβ − Iα)f, α = β;

−∫ 1

0

f(rt)tα ln tdt, α = β.

Theorem 3.1 (General solution of biharmonic equation). The general solution ofthe biharmonic equation ��u(r) = 0 in Ω∗ has the form

u(r) = v(r) +1

4r2I1/2w(r), r ∈ Ω∗, (3.1)

where v, w are arbitrary harmonic functions in Ω∗, and �u = w.

Proof. Let v, w be harmonic functions in Ω∗ and u is expressed by the formula(3.1).

Then with the help of the properties of the operator Iα we will find

�u =1

4�r2I1/2w =

3

2I1/2w + r · ∇I1/2w =

3

2I1/2w + w − 3

2I1/2w = w, (3.2)

i.e., u is biharmonic. Conversely, let u be the biharmonic function. Let w = �u,and let

v = u− 1

4r2I1/2�u.

Then the function u is represented in the form (3.1) in terms of two harmonicfunctions, the function w is harmonic by virtue of (3.2), and �u = w is also byvirtue of (3.2). �

With the help of Theorem 3.1 and the properties of the operator Iα thefollowing two statements are proved:

Theorem 3.2. The general solution of the system{∇× u(r) = v(r),

∇ · u(r) = v0(r),


where v0, v are the given harmonic respectively scalar and vector functions in Ω∗,and ∇ · v = 0, has the form

u(r) = −r× I1v +∇[r24I1/2

(v0 − r · I2∇× v

)]+∇g0, r ∈ Ω∗,

where g0 is an arbitrary harmonic function in Ω∗.

Theorem 3.3 (General solution of MTS). The general solution of MTS in Ω∗ hasthe form

f(r) = ϕ0 + r× I1∇ϕ0 +∇ψ0, r ∈ Ω∗,

where ϕ0(r), ψ0(r) are arbitrary harmonic functions in Ω∗.

We will consider in Ω∗ the Volterra integral equation of the second kind

(1 + aIα + bIβ)u(r) = f(r). (3.3)

Parameters a, b, α, β ∈ C of this equation are used to make a quadratic equation

p2 − p(a+ b+ α+ β) + aβ + bα+ αβ = 0, p ∈ C,

whose roots are

p1,2 =1

2(a+ b+ α+ β)± 1

2[(a− b+ α− β)2 + 4ab]1/2.

Then with the help of the operator Iα properties it can be proved the next

Theorem 3.4. Let p1 = p2 then the equation (3.3) has the unique solution that isexpressed in the form:

u = f + (p2 − p1)−1[(α− p1)(β − p1)I

p1 − (α− p2)(β − p2)Ip2 ]f. (3.4)

Radial integration operators are used in mathematical physics, for example,for proving of S.L. Sobolev embedding theorems, in the theory of special functions,etc. In order to solve the biharmonic equation in the theory of elasticity, they wereused in [2], [3], [12]. In the theory of quaternionic functions the operator Iα hasbeen used in the work [35] to construct a regular function with a given scalarpart. The systematic use of radial integration operators for the theory of Moisil–Theodoresco system was introduced in [16]– [20], [31].

4. A primitive function

According to [28] let us call a H-valued function F as a primitive of the regularfunction f if ∇F = f , i.e.,

∇ ·F = −f0, ∇F0 +∇× F = f . (4.1)

Obviously,∇(∇F) = −ΔF = ∇f = 0 and F is a harmonic function. Such primitivefunction is not regular.

150 Yu. Grigor’ev

Theorem 4.1 (A general representation for the primitive function). A generalrepresentation for the primitive F of a regular function f in Ω∗ has the form

F (r) = χ0 + r× I1(∇χ0 − f)−∇(1

2r2I1/2I1f0

)+∇g0, ∀r ∈ Ω∗, (4.2)

where χ0(r), g0(r) are arbitrary real-valued harmonic functions in Ω∗.

Proof. Let f be a regular function in Ω∗ and F be its primitive. We will showthat there will be found harmonic functions χ0, g0 such that F is represented inthe form (4.2). Really, we will put χ0 = F0. Then from (4.1) we will get that Fsatisfies the system

∇ · F = −f0, ∇× F = f −∇χ0, ∀r ∈ Ω∗,

where ∇ · (f − ∇χ0) = 0. Therefore, according to Theorem 3.2, F is representedin the form

F = r× I1(∇χ0 − f)− 1

4∇{r2I1/2

[f0 + r · I2∇× f

]}+∇g0,

where g0 is an arbitrary real-valued harmonic function in Ω∗. Transforming thisexpression with taking into account the regularity f and with the help of anoperator Iα properties, we will get

F = r× I1(∇χ0 − f)− 14∇

{r2I1/2

[f0 − (r · ∇)I1f0

]}+∇g0

= r× I1(∇χ0 − f)− 12∇

(r2I1/2I1f0

)+∇g0.

(4.3)

Thereby, F is represented in the form (4.2).Conversely, let f be regular in Ω∗ and F is defined by the expression (4.2),

where χ0, g0 are arbitrary harmonic functions in Ω∗. Therefore, by direct differen-tiation and by means of operator Iα properties we have the expression

∇ ·F = −f0, ∇F0 +∇× F = f ,

i.e., F is the primitive of f . �

A concept of primitive function used by us is introduced as a solution ofthe inhomogeneous MTS and differs from those mentioned in [10], [13] becausea concept of hyperdifferentiation is not used and it is not a regular (monogenic)function. In the complex analysis solutions of inhomogeneous CRS dF/dz = f areexpressed by the Theodoresco operator (transform) [39]. Therefore, the formula(4.2) can be considered as a variant of the generalized Theodoresco transform forthe MTS in a star shaped region without the use of a weakly singular integral.In [10] a brief but informative review of the different approaches to the introductionof the concept of primitive function is given, in it the inconvenience for numericalcalculations of the integral Theodoresco operator due to the presence of weaksingularity is emphasized. Particular attention is given to the derivation of explicitformulas for the primitives of monogenic polynomials. The representation (4.2)allowed us to get in the present paper the quaternion representation of the general


solution of the Lame equation different from the known ones. In addition, it isknown the quaternionic integral operators (Cauchy, Theodoresco and Bergman)allow to express the solutions of many three-dimensional boundary value problemsin closed forms [21]. Therefore it is possible the representation (4.2) will be usefulin this area.

5. Three-dimensional quaternionic analogue of theKolosov–Muskhelishvili formulae

Theorem 5.1. The general solution of the Lame equation (2.4) in Ω∗ is expressedin terms of two regular in Ω∗ functions ϕ, ψ in the form

2μu(r) = κΦ(r) − rϕ(r) − ψ(r), κ = −3λ+ 7μ

λ+ μ, (5.1)

where as Φ one can take any primitive of function ϕ, having subordinated ψ to thecondition κΦ0 = r · ϕ+ ψ0.

Proof. Let u be a solution of the Lame equation in Ω∗. We will show that therewill be found regular functions ϕ, ψ and the primitive Φ of function ϕ, and κΦ0 =r ·ϕ+ψ0 such that u is expressed in the form (5.1). We will introduce the regularfunction f = f0 + f = (λ+ 2μ)∇ · u− μ∇× u. Then the function

F = f +λ+ μ

2(λ+ 2μ)r× (∇f0)−∇ψ0, (5.2)

where ψ0 is a harmonic in Ω∗ function, will be a vector part of regular in Ω∗

function, since

∇ ·F = ∇ · f + λ+ μ

2(λ+ 2μ)∇ · [r× (∇f0)]−Δψ0 = 0,

∇× F = ∇× f +λ+ μ

2(λ+ 2μ)∇ · [r× (∇f0)]

= −∇f0 +λ+ μ

2(λ+ 2μ)[∇f0 + (r · ∇)∇f0 − 3∇f0]

= −∇ λ+ μ

2(λ+ 2μ)

[3λ+ 5μ

λ+ μf0 + (r · ∇)f0

]= −∇F0,

where

F0 =λ+ μ

2(λ+ 2μ)

[3λ+ 5μ

λ+ μf0 + (r · ∇)f0

]. (5.3)

Consequently, F = F0 + F is regular.

We will introduce another regular function

ϕ(r) = IγF (r), γ =2(λ+ 2μ)

λ+ μ> 2, (5.4)

152 Yu. Grigor’ev

from this we will find with the help of the operator Iα properties

ϕ0 = IγF0 =λ+ μ

2(λ+ 2μ)λ+ μ

[3λ+ 5μ

λ+ μIγf0 + (r · ∇)Iγf0

]=

λ+ μ

2(λ+ 2μ)

[(3λ+ 5μ

λ+ μ− γ − 1

)Iγf0 + f0

]=

λ+ μ

2(λ+ 2μ)f0,

(5.5)

and

F = (γ + 1 + r · ∇)ϕ. (5.6)

Now we will introduce the primitive Φ of the function ϕ such that κΦ0 =r ·ϕ+ψ0 (it is easy to show that such the primitive exists, see Theorem 4.1), andwe will show that the function ψ, defined by the relation

ψ(r) = κΦ(r) − rϕ(r) − 2μu(r), κ = −3λ+ 7μ

λ+ μ, ∀r ∈ Ω∗, (5.7)

i.e.,ψ0 = κΦ0 − r ·ϕ,ψ = κΦ+ rϕ0 − r×ϕ+ 2μu,

(5.8)

will be regular in Ω∗. Indeed, recalling the relations (λ+ 2μ)∇ · u = f0, (5.5) andthat ϕ ∈ R(Ω∗), we have

∇ · ψ = − κ∇ ·Φ+∇ · (rϕ0 − r×ϕ) + 2μ∇ · u = κϕ0 + 3ϕ0 + (r · ∇)ϕ0

+ r · (∇×ϕ) +2μ

λ+ 2μf0 = (κ + 3)ϕ0 + r · (∇ϕ0)− r · (∇ϕ0)

+4μ

λ+ μϕ0 =

(−3λ+ 7μ

λ+ μ+ 3 +

4μ

λμ

)ϕ0 = 0,

similarly we find

∇×ψ = −κ∇×Φ+∇× (rϕ0 − r×ϕ) + 2μ∇× u

= −κ(ϕ−∇Φ0)− r× (∇ϕ0)− [ϕ− (r · ∇)ϕ− 3ϕ]− 2f

= (2− κ)ϕ+ κ∇Φ0 − r× (∇ϕ0) + (r · ∇)ϕ− 2f ,

but, in virtue of (5.2), (5.5) and (5.6)

f = (γ + 1 + r · ∇)ϕ− r× (∇ϕ0) +∇ψ0,

and from (5.8) we will find

κ∇Φ0 = ∇ψ0 +∇(r ·ϕ) = ∇ψ0 +ϕ+ (r · ∇)ϕ+ r× (∇×ϕ)

= ∇ψ0 +ϕ+ (r · ∇)ϕ− r× (∇ϕ0),

hence

∇×ψ = (2− κ)ϕ +∇ψ0 + (r · ∇)ϕ− r× (∇ϕ0)− r× (∇ϕ0) + (r · ∇)ϕ

− 2(γ + 1)ϕ− 2(r · ∇)ϕ− 2r× (∇ϕ0)− 2∇ψ0 = −∇ψ0.

Thus, indeed ψ is regular in Ω∗, and u is represented in the form (5.1).


Conversely, let u be defined by (5.1), where ϕ, ψ ∈ R(Ω∗) are arbitrary andΦ is the primitive of ϕ, and κΦ0 = r · ϕ + ψ0. Then by direct differentiation weascertain that u satisfies the Lame equation. �

Now a general solution of the Lame equation in Ω∗ is given in terms of tworegular functions obtained by V.V. Naumov (see [17]).

Theorem 5.2. A general solution of the Lame equation (2.4) in Ω∗ is expressed interms of regular in Ω∗ two functions f,∇g0 in the form

u(r) =r

μ× I1f +∇

{r2[

3λ+ 7μ

4μ(λ+ 2μ)I1/2 − 1

μI1]f0

}+∇g0, (5.9)

and f = (λ+ 2μ)∇ · u− μ∇× u.

Proof. Let u be a solution of the Lame equation in Ω∗. We will show that therewill be found regular functions f,∇g0, such that u is expressed in the form (5.9),and f = (λ+ 2μ)∇ · u− μ∇× u. Indeed, we will put

f0 = (λ + 2μ)∇ · u, f = −μ∇× u,

in this case, as it was mentioned above f is regular. Then, according to Theorem3.2, u is determined from the above system in the form

u =r

μ× I1f +∇

[r2

4I1/2

(1

λ+ 2μf0 +

r

μ· I2∇× f

)]+∇g0, (5.10)

where g0 is a harmonic in Ω∗ function. Transforming this expression, taking intoaccount the regularity of f and using the operator Iα properties, we get

u =r

μ× I1f +∇

{r2

4

[1

λ+ 2μI1/2 − 1

μI1/2(r · ∇)I1

]f0

}+∇g0

=r

μ× I1f +∇

{r2

4

[1

λ+ 2μI1/2f0 − 1

μI1/2(f0 − 2I1f0

]}+∇g0

=r

μ× I1f +∇

{r2

4

[− λ+ μ

μ(λ+ 2μ)I1/2f0 +

4

μ

(I1/2 − I1

)f0

]}+∇g0

=r

μ× I1f +∇

{r2[

3λ+ 7μ

4μ(λ+ 2μ)I1/2 − 1

μI1]f0

}+∇g0.

Conversely, let u be defined by (5.9), where f,∇g0 are arbitrary regular functions.Then by direct differentiation we ascertain that u satisfies the Lame equation. �

General solutions (5.1) and (5.9) in Ω∗ are equivalent, and if in (5.1) oneintroduces a regular in Ω∗ function f according to the relation

f = γϕ0 + (γ + 1 + r · ∇)ϕ− r× (∇ϕ0) + ψ0, γ =2(λ+ 2μ)

λ+ μ,

154 Yu. Grigor’ev

then (5.1) turns into (5.9) and inversely: if in (5.9) one introduces regular in Ω∗

functions ϕ and ψ according to the relations

ϕ =1

γf0 + Iγ

[f +

1

γr× (∇f0)−∇ψ0

], ψ = ψ0 + r× I1∇ψ0,

where ψ0 is an arbitrary harmonic in Ω∗ function, and the primitive Φ of thefunction ϕ such that κΦ0 = r ·ϕ+ ψ0, then (5.9) turns into (5.1).

6. Plane and axially symmetric deformations

In this section it is shown that general solutions (5.1) and (5.9) of the Lameequation in Ω∗, expressed in terms of two regular functions, in the case of planedeformation goes into the general Kolosov–Muskhelishvili solution of the equationsof the plane theory of elasticity, expressed in terms of two analytic functions ofcomplex variable [29]. Also it is shown that in the case of axially symmetric defor-mation both quaternion representations go into the general Yu.I. Solov’ev solutionof equations of the axially symmetric theory of elasticity, expressed in terms oftwo generalized analytic by Vekua functions of complex variable [1].

Theorem 6.1. In the case of plane deformation ux = ux(x, y), uy = uy(x, y),uz = 0 the quaternion representation (5.9) of the general solution of the Lameequation goes into the general Kolosov–Muskhelishvili solution of the equations ofthe plane theory of elasticity

2μ(ux + iuy) = κϕ(ζ) − ζϕ′(ζ) − ψ(ζ), κ =λ+ 3μ

2(λ+ μ)= 3− 4ν, (6.1)

where ϕ(ζ), ψ(ζ) are the analytic functions of the complex variable ζ = x+ iy.

Proof. In the considered case of plane deformation, when ux = ux(x, y), uy =uy(x, y), uz = 0, from the relations(2.5) we have fx = fy = 0, f0 = f0(x, y), fz =fz(x, y), and, as it was noted in the introduction, the function f(ζ) = f0(x, y) −ifz(x, y) will be the analytic function of complex variable ζ = x+ iy, also⎧⎪⎪⎨⎪⎪⎩

ux,x + uy,y =1

λ+ 2μf0,

uy,x + ux,y = −fzμ.

We will write the condition uz = 0 for the representation (5.9):

uz = 2z

[3λ+ 7μ

4μ(λ+ 2μ)I1/2 − 1

μI1]f0 + g0,z = 0.

Integrating this equation over z, we will get

z2[

3λ+ 7μ

4μ(λ+ 2μ)I1/2 − 1

μI1]f0 + g0 = ψ0(x, y), (6.2)


where ψ0(x, y) is so far an arbitrary function of integration. Since f0, g0 areharmonic, then from (6.2) we get

ΔΔψ0(x, y) = 0,

Δψ0(x, y) = 2

[3λ+ 7μ

4μ(λ+ 2μ)I1/2 − 1

μI1]f0(x, y).

(6.3)

Consequently, under Theorem 3.1, the general solution of the problem (6.3) hasthe form

ψ0(x, y) = χ0(x, y) +x2 + y2

2I0[

3λ+ 7μ

4μ(λ+ 2μ)I1/2 − 1

μI1]f0, (6.4)

where χ0(x, y) is an arbitrary harmonic function. Now from (6.2), taking intoaccount (6.4), we find

g0 =

(−z2 + x2 + y2

2I0)[

3λ+ 7μ

4μ(λ+ 2μ)I1/2 − 1

μI1]f0(x, y) + χ0(x, y). (6.5)

Substituting (6.5) into (5.9) for ux, we will get

ux =y

μI1fz +

1

2μ

{(x2 + y2)

[λ+ 3μ

2(λ+ 2μ)I0 − I1

]f0

},x

+ χ0,x.

Similarly, we will get

uy =x

μI1fz +

1

2μ

{(x2 + y2)

[λ+ 3μ

2(λ+ 2μ)I0 − I1

]f0

},y

+ χ0,y.

We will introduce a new analytic function ϕ(ζ), such that

ϕ′(ζ) = ϕx,x(x, y) + iϕx,y(x, y) =

=λ+ μ

2(λ+ 2μ)f(ζ) =

λ+ μ

2(λ+ 2μ)[f0(x, y) − ifz(x, y)] .

(6.6)

Then, making transformations, for ux and uy we will get the expressions

2μux = κϕx − (xϕx,x + yϕy,x)− κϕx(0) + 2μχ0,x,

2μuy = κϕy − (yϕx,x − xϕy,x)− κϕy(0) + 2μχ0,y;

κ =λ+ 3μ

λ+ μ= −3− 4ν.

(6.7)

Introducing another analytic function

ψ(ζ) = 2μ (−χ0,x + iχ0,y) + κ [ϕx(0)− iϕy(0)] ,

from (6.7) we get (6.1). �

Theorem 6.2. In the case of axially symmetric deformation the quaternion repre-sentation (5.9) of the general Lame equation goes into the general Yu.I. Solov’evsolution of equations of the axially symmetric theory of elasticity, expressed interms of two generalized analytic by Vekue functions of complex variable [1, p. 296].

156 Yu. Grigor’ev

Proof. We write the representation (5.9) in the cylindric system of coordinates�, ϕ, z (x = � cosϕ, y = � sinϕ, z = z):

u = − z

μI1fϕ +

{(�2 + z2)

[3λ+ 7μ

4μ(λ+ 2μ)I1/2 − 1

μI1]f0

},�

+ g0,�,

v =1

μ

(zI1f� − �I1fz

)+

1

�

{(�2 + z2)

[3λ+ 7μ

4μ(λ+ 2μ)I1/2 − 1

μI1]f0

},ϕ

+1

�g0,ϕ,

w =�

μI1fϕ +

{(�2 + z2)

[3λ+ 7μ

4μ(λ+ 2μ)I1/2 − 1

μI1]f0

},z

+ g0,z. (6.8)

Here (u, v, w) and (f�, fϕ, fz) are the physical projections of u f respectively.We will consider the axially symmetric deformation, when u = u(z, �), w =w(z, �), v = 0. Then from (2.3) it follows that

f� = fz = 0, f0 = f0(z, �), fϕ = fϕ(z, �) (6.9)

and, as it was noted in the introduction, the function f(ζ) = f0(z, �) − ifϕ(z, �)will be the generalized by Vekua function of complex variable ζ = z+ i�. From thecondition v = 0, from (6.8) and (6.9) it follows that g0,ϕ = 0, i.e., g0 is the axiallysymmetric harmonic function:

Δg0 =1

�(�g0,�),� + g0,zz = 0. (6.10)

It is easy to show that the function F (ζ) = Fz+iF� = g0,z−ig0,� will be generalizedanalytic. Indeed, in virtue of (6.10) we have

Fz,z = g0,zz = −1

�(�g0,�),� =

1

�(�F�),

on the other hand

F�,z = g0,�z = −F�,z,

i.e., the generalized Cauchy–Riemann conditions for F (ζ) are satisfied:

Fz,z =1

�(�F�),�, Fz,� = −F�,z.

We will introduce a new generalized analytic function ϕ(ζ), such that

ϕ′(ζ) = ϕz,z(z, �) + iϕ�,z(z, �) =

=λ+ μ

2(λ+ 2μ)f(ζ) =

λ+ μ

2(λ+ 2μ)[f0(z, �)− ifϕ(z, �)] ,

i.e., the generalized Cauchy–Riemann conditions for ϕ(ζ) are satisfied:

ϕz,z =1

�(�ϕ�),�, ϕz,� = −ϕ�,z.


We will note that if ϕ(ζ) is generalized analytic, then ϕ′(ζ) = ∂zϕ(ζ) will also begeneralized analytic. Taking into account all this, from (6.8) we have

2μw = − 4(λ+ 2μ)

λ+ μ�I1ϕ�,z +

2(3λ+ 7μ)

λ+ μzI1/2ϕz,z − 8(λ+ 2μ)

λ+ μzI1ϕz,z (6.11)

+3λ+ 7μ

λ+ μ(�2 + z2)I3/2ϕz,zz − 4(λ+ 2μ)

λ+ μ(�2 + z2)I2ϕz,zz + 2μg0,z.

Then it can be shown that

2μw = κϕz − (zϕz,z + �ϕ�,z)− κ

2I−1/2ϕz + 2μg0,z, (6.12)

where

κ =3λ+ 7μ

2(λ+ μ).

In an analogical way we also have

2μu = κϕ� − (�ϕz,z − zϕ�,z) +κ

2I−1/2ϕ� + 2μg0,�. (6.13)

Introducing another generalized analytic function

Ψ(ζ) = 2μ (−g0,z + ig0,�) +κ

2I−1/2ϕ(ζ),

from (6.12) and (6.13) we will finally get the general Yu.I. Solov’ev solution ofequations of the axially symmetric theory of elasticity

2μ(w + iu) = κϕ(ζ) − ζϕ′(ζ)−Ψ(ζ), (6.14)

where ϕ(ζ), Ψ(ζ) are the generalized analytic by Vekua functions of the complexvariable ζ = z + i�. �

7. Normal loading of elastic sphere

In this section from the quaternion representation (5.9) a new representation of thegeneral solution of the Lame equation in Ω∗ is obtained expressed in terms of threeharmonic functions. With the help of that representation the solution of 2nd basicproblem of the theory of elasticity for the sphere, when on the boundary of theball a purely normal self-balanced load is applied, is obtained in the closed form.The solution of the problem is also expressed through a solution of the Dirichletproblem for the function, which is harmonic in the sphere.

7.1. General solution

Using Theorem 3.3 from the quaternion representation (5.9) it can be proved

Theorem 7.1. The general solution of the Lame equation (2.4) in Ω∗ is expressedin terms of three arbitrary harmonic in Ω∗ functions χ0, ψ0, g0 in the form

μu(r) = rI0χ0 +∇{r2[

3λ+ 7μ

4(λ+ 2μ)I1/2 − I0

]χ0

}+ r× (∇ψ0) +∇g0, (7.1)

and ∇ · u = (1/(λ+ 2μ))χ0.

158 Yu. Grigor’ev

The components of stress tensor for this general solution in the sphericalcoordinates r, θ, ϕ have the forms:

σrr =5

8(1− ν)f − r

4− 4νf,r − τ

2I1/2f + 2g,rr,

σrθ =

[(I0 − τI1/2 − 1

4− 4ν

)f +

2

r2(rg,r − g)

],θ

+1

r sin θ(ψ − rψ,r),φ ,

σrϕ =1

sin θ

[(I0 − τI1/2 − 1

4− 4ν

)f +

2

r2(rg,r − g)

],ϕ

− 1

r(ψ − rψ,r),θ ,

(7.2)

here

τ =7− 8ν

8(1− ν), ν =

λ

2(λ+ μ),

ν is Poisson’s ratio.

In the paper [12] it is shown that in a star shaped region in the Papkovich–Neuber representation without loss of generality one can set equal to zero oneharmonic function, but only under certain restrictions on Poisson’s ratio. In con-trast to this our representation (7.1) by means of three harmonic functions is thegeneral solution without any restrictions on the elastic constants.

7.2. The method of solution

By means the representation (7.1) all main problems of an equilibrium of an elasticsphere can be solved in the closed forms. We will denote through U and S thesphere of a radius R and its boundary. For example we will consider the 2ndbasic problem of equilibrium of elastic sphere with a purely normal load on theboundary: {

Lu(r) = 0, r ∈ U, C2(U) ∩ C1(U)

σrr|S = σ(θ, ϕ) ∈ C0(S); σrθ|S = σrϕ|S = 0.(7.3)

Let the condition of the main vector of applied external forces being equal to zerobe satisfied ∮

S

σdS =1

R

∮S

σ(θ, ϕ)rdSr = 0, (7.4)

the principal moment for a purely normal load is always equal to zero. We willimpose a stronger limitation

σrr ∈ C0(U), σrθ ∈ C1(U), σrϕ ∈ C1(U).

The solution of the problem (7.3), (7.4) we will find in the form (7.1), wherethe harmonic functions have the properties

f ∈ C2(U), g, ψ ∈ C3(U). (7.5)

In virtue of (7.5) from (7.2) it is seen that from the zero boundary conditions forσrϑ, σrϕ it follows that ψ−rψ,r ≡ Ψ(r) is harmonic and Ψ = C1 = const in U , and


the functions f and g for r = R on the boundary S of the sphere U are related bythe relation:

2

R2(g − rg,r)

∣∣∣∣r=R

−[(

I0 − τI1/2 − 1

4− 4ν

)f

]∣∣∣∣r=R

= C2 = const, (7.6)

but in (7.6) in virtue of (7.5) and the operator Iα properties the functions in thesquare brackets and parentheses will be regular harmonic in the sphere U . Hence,in virtue of the uniqueness of the solution of the Dirichlet problem from (7.6) weget the relation, which is true everywhere in the sphere U :

g − rg,r =R2

2

[(I0 − τI1/2 − 1

4− 4ν

)f + C2

], ∀r ∈ U. (7.7)

Differentiating (7.7) with respect to r, we will find

g,rr = −R2

2r

[(I0 − τI1/2 − 1

4− 4ν

)f

],r

, (7.8)

substituting (7.8) into the expression for σrr from (7.2), we will obtain that σrr isexpressed only through f :

σrr =5

8(1− ν)f − r

4− 4νf,r− τ

2I1/2f − R2

r

[(I0 − τI1/2 − 1

4− 4ν

)f

],r

, (7.9)

hereof and from (7.3) we will get the boundary condition

σrr|r=R =

[5

8(1− ν)f − τ

2I1/2f

]∣∣∣∣r=R

−R(I0f − τI1/2f

),r

∣∣∣∣r=R

= σ(ϑ, ϕ).

(7.10)We will clarify the structure of f . In follows from the condition of solvability

(7.4) that the expansion of the regular in the sphere U harmonic function f intoa series of spherical functions has the form

f = f0 + f2, f0 = f(0), f2 =∞∑

n=2

n∑m=−n

anm

( r

R

)nY mn (ϑ, ϕ).

We will introduce a new harmonic function F :

F (r) =

[1

2(1− ν)− 2τI1/2 + I0

]f. (7.11)

It is clear that the structure of the function F is analogous to the structure of thefunction f . With the help of the operator Iα properties one can show that theboundary condition (7.10) can be written in the form

F |r=R = σ(ϑ, ϕ).

Thereby, for the new harmonic function we got the Dirichlet problem{ΔF (r) = 0, r ∈ U,

F |r=R = σ(ϑ, ϕ).(7.12)

160 Yu. Grigor’ev

With the help of Theorem 3.3 from (7.11) we find

f = 2(1− ν) (F + 2ReAIωF ) ,

ω =1

2(−1 + 2ν + iq), q =

√3− 4ν2,

A =1

4

[3− 4ν +

i

q(8ν2 − 6ν − 1)

].

(7.13)

Now we will consider the equation (7.7). The homogeneous equation (7.7)g − rg,r = 0 has a nontrivial solution g0 = r · a, where a is an arbitrary constant.This solution, according to (7.1), defines a rigid displacement u = a. A particularsolution for it with the constant right-hand side is also a constant and it does notcontribute to the solution of the original problem. Discarding these solutions, wewill leave only the particular solution with the right-hand side, formed from f2,which has the form

g = g2 =R2

2I−2

(τI1/2 − I0 +

1

4− 4ν

)f2.

Substituting here (7.13) and transforming the obtained expression we will get

g2 =R2

6

(I−2 + 2ReBIω

)F2,

B =1

4

[1 +

i

q(9 − 10ν)

].

(7.14)

We will consider the equation for ψ: ψ − rψ,r = C1. Its solution has the formψ = C1 + r · b, where b is an arbitrary constant vector. Therefore the function ψin virtue of (7.1) defines the solution in the form of rigid rotation u = r× b.

Finally, substituting (7.13) and (7.14) in (7.1) and carrying out some trans-formations we have the result

Theorem 7.2. The solution of the problem (7.3)–(7.4) exists, is defined up to arigid displacement and has the form

μu(r) = 2rRe(AIωF )− r2∇Re(BIωF ) +R2

6∇ (I−2 + 2ReBIω

)F2, (7.15)

where

ω =1

2(−1 + 2ν + iq); q =

√3− 4ν2,

A =1

4

[3− 4ν +

i

q(−1− 6ν + 8ν2)

],

B =1

4

[1 +

i

q(9− 10ν)

],

F2 = F − F (0)− r limr→0

∂

∂rF,

(7.16)

the function F is the solution of the Dirichlet problem (7.12).


7.3. Closed form of the solution

Next, we will obtain the solution of the original problem in quadratures. For thisone needs to substitute into (7.15) the expression for the function F in the formof the Poisson integral, which is written in the form

1

4πR2

∮σ1− t2

s3dS, s2 = 1− 2tc+ t2, t = r/R, (7.17)

where

c = cos γ = cos θ cos θ′ + sin θ sin θ′ cos (ϕ− ϕ′); dS = R2 sin θ′dθ′dϕ′,

∮ΦdS =

∮S

Φ(r; r′)dSr′ = R2

2π∫0

dϕ′π∫

0

dθ′ sin θ′Φ(r;R, θ′, ϕ′).

We will note that in virtue of the equilibrium conditions (7.4) we will have therelation

∮σcdS = 0. For calculating I−2F2 the integral is necessary

I−2

(1− t2

s3− 1− 3ct

)= 1− 5ct+

2

s− 3s− 3ct ln

ζ

2, ζ = 1− ct+ s.

Substituting (7.17) into (7.15), taking into account the noted above considerations,using the formulas, which are easy to verify according to the operator Iα properties:

Iω(1− t2

)s−3 = 2s−1 − (2ω + 1)Iωs−1, Reω > −1;

t∂tIω(s−α − 1

)= s−α − 1− (ω + 1)Iω

(s−α − 1

), Reω > −2,

introducing new complex variables A1 = −2(1 + 2ω)[2A + (1 + ω)B], 3A2 =2(1 + 2ω)(1 + ω)B, A3 = 2(1 + 2ω)(1 + ω)B we have the next

Theorem 7.3. The solution of the problem (7.3)–(7.4) up to a rigid displacementhas the form:

ur(r) =1

8πμR

∮σ

{2(1− ν)

1 + t2

ts+

(1 − t2)2

2ts3− s

t

− c ln ζ +1

tRe

[(A1t

2 +A2)Iω 1

s

]}dS;

(7.18)

{uθ(r)

uϕ(r)

}=

1

8πμR

∮σ

{ξ

η

}[1− t2

s3+

1 + s+ tcs

ζs

− ln ζ +

(t2 − 1

3

)Re

(A3I

ω+1 1

s3

)]dS,

162 Yu. Grigor’ev

where r ∈ U ;

ζ = 1− tc+ s, ω =1

2(−1 + 2ν + iq), q =

√3− 4ν2;

A1 = 2− 6ν + 4ν2 − i

q(3 + ν − 12ν2 + 8ν3);

A2 = −1− 2ν − 4ν2 − i

q(2− 5ν − 4ν2 + 8ν3);

A3 =3

2

[−3 + 4ν +

i

q(1 + 6ν − 8ν2)

].

(7.19)

We will note that the here appearing integrals of the form Iβs−α are expressedin terms of the Appell hypergeometric function

Iωs−α =1

ω + 1F1

(ω + 1,

α

2,α

2, ω + 2, teiγ , te−iγ

), Re ω > −2, (7.20)

where α = const ∈ C; t ∈ [0, 1).

7.4. Case of axial symmetry

We will consider a particular case of axial symmetry, when in the boundary condi-tion σ = σ(ϑ). The integration over dϕ′ in the formulas (7.18) gives the followingexpressions for the displacement components (uϕ = 0):

ur =R

2πμ

π∫0

dϑ′ sinϑ′σ(ϑ′)[1− 2ν

1 + ν

πt

2+ 2(1− ν)

1 + t2

tU(t)

+1− t2

2t(2t∂tU(t) + U(t)) +

1

tRe

1∫0

(Pt2 +Q

y1+n1+

1

y2

)U(ty)dy

⎤⎦ ,

uθ =R

2πμ∂ϑ

π∫0

dϑ′ sinϑ′σ(ϑ′)[(1− t2)U(t)

+Re

1∫0

(St2 + T

y1+n1+

1

y2

)U(ty)dy

⎤⎦ ,

(7.21)

where

n1 + 1 = −ω, P = A1, Q = A2, S = A3, T = −1

3S,

i.e.,

n1 =1

2(−1− 2ν + iq),

√3− 4ν2,

P = 2− 6ν + 4ν2 +i

q(3 + ν − 12ν2 + 8ν3),

Q = −1− 2ν − 4ν2 +i

q(2− 5ν − 4ν2 + 8ν3),


S =3

2

[−3 + 4ν − i

q(1 + 6ν − 8ν2)

],

T =1

2

[3− 4ν +

i

q(1 + 6ν − 8ν2)

]. (7.22)

We use notation

U(t) ≡ U(t, ϑ, ϑ′) =1

hK(k)− π

2(1 + t cosϑ cosϑ′).

where K(k) is the complete normal Legendre elliptic integral of the first kind.As it was expected, the expressions (7.21) coincide with the known result [8].

We will note that in the paper [8] there are given the wrong values for constantsS and T , the right values for them are given here in (7.22).

We note that the method of solving the problem for the ball given hereappeared only when using quaternion functions. An another more general approachfor solving boundary problems of mathematical physics by means of the theory ofquaternion functions are given in [22].

8. Conclusions

In this paper the theory of Moisil–Theodoresco system in terms of regular quater-nionic functions of reduced quaternion variable is used. A radial integration tech-nique is used systematically for solving the arising problems. Unlike [4], we usedanother notion of primitive of regular function. Therefore, we have proved thenew version of three-dimensional quaternionic analogue of the complex Kolosov–Muskhelishvili formulae. Also another quaternion representation of Lame equationsolution is presented. An equivalence of these two representations is shown. It isshown that the quaternion representation in the case of plane deformation goesinto the general Kolosov–Muskhelishvili solution and in the case of axially sym-metric deformation goes into the general Yu.I. Solov’ev solution. As applicationsthe problem of an elastic sphere equilibrium in the case of normal loading is solved.Using the special representation of the solution of the Lame equation, which is aconsequence of quaternion representation, the solution is expressed in terms of oneharmonic function, which is the solution of the Dirichlet problem with the bound-ary condition as the original problem. This solution is also expressed in terms ofquadratures of elementary functions and Appell hypergeometric function. In theparticular case of axial symmetry the coincidence with the known result is ob-tained. In the general case it can be shown that all main problems of an elasticsphere equilibrium by means of the proposed method may be expressed in termsof three independent harmonic functions.

164 Yu. Grigor’ev

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[26] V.G. Maz’ya and V.D. Sapozhnikova,A note on regularization of a singular system ofisotropic elasticity. Vestnik Leningr. univers. Ser. mat., mekh. i astron [in Russian],V. 7, N. 2 (1964), Leningr. gos. univers., Leningrad, 165–167.

[27] P. Mel’nichenko and E.M. Pik, Quaternion equations and hypercomplex potentials inthe mechanics of a continuous medium. Soviet Applied Mechanics, Volume 9, Issue4 (April 1973), 383–387. DOI 10.1007/BF00882648.

[28] M. Misicu, Representarea ecuatilor echilibrului elastic prin functii monogene de cu-aterninoni. Bull. Stiint. Acad. RPR. Sect. st. mat. fiz., V. 9, N. 2 (1957), 457–470.

[29] N.I. Muskhelishvili, Some Basic Problems of the Mathematical Theory of Elasticity.Springer, 1977.

[30] V.V. Naumov and Yu.M. Grigor’ev, The Laurent series for the Moisil–Theodorescosystem. Dynamics of Continuous Medium [in Russian], no. 54 (1982), Inst. Gidrodin.Sib. Otd. Akad. Nauk SSSR, Novosibirsk, 115–126.

[31] V.V. Naumov, Solution of two main problems of an equilibrium of an elastic spherein a closed form. Dynamics of Continuous Medium [in Russian], no. 54 (1986), Inst.Gidrodin. Sib. Otd. Akad. Nauk SSSR, Novosibirsk, 96–108.

[32] D.D. Penrod, An analogue of the Kolosov–Muskhelishvili formulae in three dimen-sions. Quart. of Appl. Math., V. 23, N. 4 (1966), 313–322.

166 Yu. Grigor’ev

[33] A.A. Pimenov and V.I. Pushkarev, The use of quaternions to generalize the Kolosov–Muskhelishvili method to three-dimensional problems of the theory of elasticity. Jour-nal of Applied Mathematics and Mechanics, Volume 55, Issue 3 (1991), 343–347. DOI10.1016/0021-8928(91)90036-T.

[34] G. Polozij, The Theory and Application of p-Analytic and pq-Analytical Functions[in Russian]. Naukova Dumka, Kiev, 1973.

[35] A. Sudbery, Quaternionic analysis. Mathematical Proceedings of the CambridgePhilosophical Society, no. 85 (1979), 199–225. DOI:10.1017/S0305004100055638.

[36] W. Sproßig and K. Gurlebeck, A hypercomplex method of calculating stresses inthree-dimensional bodies. In: Frolik, Z. (ed.): Proceedings of the 12th Winter Schoolon Abstract Analysis. Section of Topology. Rendiconti del Circolo Matematico diPalermo, Serie II, Supplemento no. 6 (1984), 271–284. http://dml.cz/dmlcz/701846.

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Yuri Grigor’evTheoretical Physics DepartmentNorth-Eastern Federal University58, Belinsky Str.Yakutsk, 677000 Russiae-mail: [email protected]

http://dml.cz/dmlcz/701846




On the Continuous Coupling of Finite Elementswith Holomorphic Basis Functions

Klaus Gurlebeck and Dmitrii Legatiuk

Abstract. The main goal of this paper is to improve the theoretical basis ofcoupling of an analytical and a finite element solution to the Lame–Navierequations in case of singularities caused by a crack. The main interest is toconstruct a continuous coupling between two solutions through the wholeinteraction interface. To realize this continuous coupling so-called couplingelements are introduced, which are based on a new interpolation operator. Inthe convergence study of the finite element method the invariant subspacesof the interpolation operator plays a crucial role. In previous studies it hasbeen shown that for a given special distribution of interpolation nodes thecorresponding interpolation problem is uniquely solvable. In this paper weextend the result to the case of an arbitrary number of nodes. This result isthe basis for a well-defined interpolation operator with all properties whichare needed for the convergence analysis of the coupled finite element method.

Mathematics Subject Classification (2010). Primary 74S70, 74S05, 30E05,35J25; Secondary 30J15, 35Q74, 74R99.

Keywords. Lame–Navier equations, complex function theory, analytical solu-tion, Kolosov–Muskhelishvili formulae, FEM, coupling, interpolation problem,singularity.

1. Introduction

The idea of a coupling comes from engineering problems containing different typesof singularities (like for instance cracks, gaps, corners). To handle such problems bynumerical methods (like finite element method, finite difference method, etc.) oneneeds to perform some adaptations of the numerical scheme and usually it requiresalso a refined mesh in the region near to a singularity. At present, the finite elementmethod is the most popular numerical method in computation mechanics.

The research of the second author is supported by the German Research Foundation (DFG).

Switzerland

168 K. Gurlebeck and D. Legatiuk

An alternative to numerical methods for problems of linear elasticity are themethods of complex function theory. By using the Kolosov–Muskhelishvili formulaeone can describes the near-field solution of a crack tip by only two holomorphicfunctions Φ(z) and Ψ(z), z ∈ C [8]. The analytical solution based on the complexfunction theory gives us a high accuracy of the solution in the neighbourhood ofthe singularity. Because of using exact solutions of the partial differential equationsall details of the mathematical model are preserved.

The disadvantage of the complex analytic approach is that the full linearelastic boundary value problem can be solved explicitly only for some elemen-tary (simple) or canonical domains. Domains coming from practical engineeringproblems usually are more complicated. Therefore it makes sense to ask for acombination of an analytical and a finite element solution in one procedure. Theproblem of such combination comes from a coupling of two different solutions. Onone hand, the analytical solution which is constructed by the complex functiontheory has a purely analytic form but on the other hand, the standard finite ele-ment solution is based on spline functions. Due to that fact we need to considermore carefully this coupling process.

In previous research [4, 3, 6] we have introduced the main idea of a newmethod of coupling an analytical and a numerical solution (FE-solution). Thismethod allows to get a continuous coupling between analytical and finite elementsolutions through the whole interaction interface. Usually, analytic solutions andFE-solutions are coupled only through the nodes of the mesh [9, 10]. This pointwisecoupling leads to a simple integration of the extra elements into the typical finiteelement scheme but out of the nodes the numerical solution will have jumps. Look-ing at the quality of the solution it is not completely satisfying that one improvesthe approximation of a point singularity (zero-dimensional) of the displacementfield and as a result the displacement field has a one-dimensional jump.

To overcome this problem we construct a special element that contains anexact solution to the differential equation with the correct singularity and so-calledcoupling elements. The request for these coupling elements is to insure C0 conti-nuity for displacements. For that reason a special interpolation operator has beenconstructed that preserves the analytical solution on the coupling interface, cou-ples it continuously with special elements which have a polynomial connection tothe standard elements. In [2] following P.G. Ciarlet [1] some basic steps for con-vergence analysis of the proposed method have been performed. In this theory oneof the most important roles is played by the unisolvence property of the inter-polation operator that is used for the finite element approximation. In [4] it hasbeen shown that for a given special distribution of interpolation nodes the cor-responding interpolation problem is uniquely solvable. In this paper these resultsare generalized to the case of an arbitrary number of nodes. This is the necessaryresult to define the basis functions for the interpolation operator which permitan arbitrary refinement of the mesh and makes the method practically applicable.Based on this result the convergence and error estimates of the proposed schemecan be proved but this not the purpose of this paper.

On the Continuous Coupling 169

To prepare the theorem about the new interpolation operator we will recallthe basic ideas of coupling from [3, 4]. For that reason we start with the analyticalsolution to a crack tip problem. After that we introduce the problem of couplingand will formulate the general interpolation theorem.

2. Geometrical settings and the analytical solution

Following [3, 4], in this section a construction of the analytical solution is discussedand the general description of a domain and its decomposition for the specialpurposes is introduced. To construct the exact solution in the crack-tip region weare going to work in the field C of one complex variable, where we identify eachpoint of the complex plane C with the ordered pair z = (x1, x2) ∈ R2, x1, x2 ∈ R

or equivalently with the complex number z = x1 + ix2 ∈ C, where i denotes theimaginary unit.

Let now Ω ⊂ C be a bounded simply connected domain containing a crack.To describe the behaviour of the continuum near the crack-tip correctly we aregoing to model more precisely the near-field domain, called ΩSE (see Figure 1).The domain ΩSE can be interpreted as a special element in the triangulation Fh

over the domain Ω. The special element is always located at the crack tip, i.e., atthe origin of a Cartesian coordinate system.

Figure 1. Geometrical setting of special element

The domain ΩSE is decomposed in two sub-domains, ΩSE = ΩA ∪ ΩD, sep-arated by the fictitious joint interface ΓAD = ΩA ∩ ΩD. The discrete “numerical”domain, denoted by ΩD, is modelled by two different kinds of elements: the CST-elements with C0(Ω) continuity (elements A–H in Figure 1) and the Coupling-elements with C0(Ω) continuity to the CST-elements, and with C∞(Ω) continuity


on the interface ΓAD (elements I–IV in Figure 1), which couple the “numerical”domain ΩD with the “analytical” domain ΩA. The C∞(Ω) continuity on the in-terface ΓAD should be understood in the sense, that the interpolation functionsare infinitely differentiable on the interface. But, this does not mean automati-cally that the connection between elements will be better than C0(Ω). For thatwe would need to introduce some additional conditions.

The nodes 3, 10 and 4, 11 belong to the upper and lower crack faces, respec-tively. We call the sub-domain ΩA analytical in that sense, that the constructedsolutions are exact solutions to the differential equation in ΩA. Analogously, thenumerical sub-domain ΩD means, that the constructed solutions are based on thefinite element approximation. The idea behind this special element is to get thecontinuous connection through the interface ΓAD by modifying the shape functionsover the curved triangles I–IV.

As a result, we introduce a triangulation Fh over the domain Ω by threefamilies of finite elements

Ω = ∪KA∈FhKA ∪KCE∈Fh

KCE ∪KCST∈FhKCST,

where the KA-element is based on the analytical solution in ΩA, KCE are theCoupling-elements and KCST are the classical CST-elements. A connection be-tween the elements KA, KCE and KCST is defined by common sets of degrees offreedom. Additionally, the connection between KA and KCE is supplemented bycontinuous connection through the interface ΓAD.

We will solve the following boundary value problem in the domain Ω:⎧⎪⎪⎪⎨⎪⎪⎪⎩−μΔu− (λ + μ)grad divu = f in Ω,u = 0 on Γ0,2∑

j=1

σij(u)vj = gi on Γ1, 1 ≤ i ≤ 2.(2.1)

System (2.1) of equations of linear elasticity in the plane describes the state ofan elastic body in the case of two dimensions and in this article we concentrateourselves to the plane strain state, i.e., u3 = 0, ε3j = 0, j = 1, . . . , 3. In Figure 1 thedomain Ω represents a volume, that is occupied by a solid body. The boundaries Γ0

and Γ1 are defined with Dirichlet and Neumann boundary conditions, respectively.Surface forces of density g and volume forces with the density f are given, udenotes the displacements, vj are components of the unit outer normal, λ and μare material constants (see, e.g., [1]).

The crack tip produces a singularity of the solution in the domain Ω. Dueto that fact, on the one hand it must be handled in a proper way to get the rightbehavior of the solution near the singularity. But on the other hand in the part ofthe domain Ω which is free of singularities one can use the standard finite elementmethod. For that reasons we will construct the analytical solution to the cracktip problem near the singularity by using complex function theory and couple thissolution with a finite element solution for the part of domain without singularity.


To construct the analytical solution to the crack-tip problem we are going towork with the Kolosov–Muskhelishvili formulae (see [8]), which are given by

2μ(u1 + i u2) = κΦ(z)− zΦ′(z)−Ψ(z),

σ11 + σ22 = 2[Φ

′(z) + Φ′(z)

],

σ22 − σ11 + 2i σ12 = 2[zΦ′′(z) + Ψ

′(z)],

(2.2)

where Φ(z) and Ψ(z), z ∈ C are two holomorphic functions. The factor κ is theKolosov constant, which is defined by

κ =

{3− 4ν for plane strain,3− ν

1 + νfor plane stress.

The crack faces are assumed to be traction free [7], i.e., the normal stressesσϕϕ and the shear stresses σrϕ on the crack faces vanish for ϕ = π or ϕ = −π,where ϕ and r are polar coordinates (see Figure 1). Corresponding to [8] theKolosov–Muskhelishvili formulae in polar coordinates read as follows:

2μ(ur + i uϕ) = e−iϕ(κΦ(z)− zΦ′(z)−Ψ(z)

)σrr + σϕϕ = 2

[Φ

′(z) + Φ′(z)

],

σϕϕ − σrr + 2i σrϕ = 2e2i ϕ[zΦ

′′(z) + Ψ

′(z)].

(2.3)

By adding the last two equations of (2.3) we get the following equation whichconnects the stresses σϕϕ and σrϕ

σϕϕ + i σrϕ = Φ′(z) + Φ′(z) + e2i ϕ[zΦ

′′(z) + Ψ

′(z)]. (2.4)

The functions Φ(z) and Ψ(z) will be written as series expansions

Φ(z) =∞∑k=0

akzλk , Ψ(z) =

∞∑k=0

bkzλk , (2.5)

where ak and bk are unknown coefficients, which should be determined throughthe boundary conditions for the global problem and the powers λk describe thebehaviour of the displacements and stresses near the crack tip and should bedetermined through the boundary conditions on the crack faces.

After substituting (2.5) into (2.4) one can calculate the powers λk and ob-tains a relation between the unknown coefficients ak and bk. Finally we have thefollowing expressions for displacements and stresses in Cartesian coordinates

2μ(u+ i v) =

∞∑n=1,3,...

rn2

[an(κ eiϕ

n2 − e−iϕn

2

)+

n

2an

(e−iϕn

2 − e−iϕ(n2 −2)

)]+

∞∑n=2,4,...

rn2

[an(κ eiϕ

n2 + e−iϕn

2

)+

n

2an

(e−iϕn

2 − e−iϕ(n2 −2)

)]. (2.6)

The displacement field (2.6) satisfies all the conditions on the crack faces. Theasymptotic behaviour at the crack tip is controlled by half-integer powers. To avoidunboundedness for (n < 0) and discontinuity (for n = 0) of the functions (2.6)at the origin the series must begin with n = 1. Now we are going to solve theproblem of coupling in a way to get a continuous displacement field through theboundary ΓAD.

3. The interpolation problem for the coupling

The purpose of this section is to define an interpolation operator on the jointinterface ΓAD and to prove the unique solvability of the corresponding interpolationproblem. How to get the desired continuous coupling through the interface is shownalready in [4] and will not be repeated here. Let us consider n nodes on the interfaceΓAD belonging to the interval [−π, π] (see Fig. 2).

Figure 2. The coupling problem

As the interpolation function fn(ϕ) we use partial sums of the analyticalsolution (2.6) restricted to the interface ΓAD (i.e., r = rA). Additionally, to beable to represent by this interpolation function all polynomials up to a certain


degree, we add a constant to our ansatz and we have

fn(ϕ) =

N1∑k=0,2,...

rk2

A

[ak

(κ eiϕ

k2 + e−iϕk

2

)+

k

2ak

(e−iϕk

2 − e−iϕ( k2−2)

)]

+

N2∑k=1,3,...

rk2

A

[ak

(κ eiϕ

k2 − e−iϕk

2

)+

k

2ak

(e−iϕk

2 − e−iϕ( k2−2)

)], (3.1)

where the number of basis functions is related to n as follows:

N1 = n− 2m, with

{m = 1 for even n,m = 0 for odd n,

N2 = n− 2m, with

{m = 0 for even n,m = 1 for odd n.

In [4] it is shown that for n = 5 the corresponding interpolation problem at thenodes on the circle can be solved for arbitrary data. Numerical experiments forthe case of a hinge were presented in [6, 11] and showed a very good performancecompared with commercial finite element software. For an exact reasoning and asa basis for the convergence of the coupled FE-method we need the solvability forarbitrary number and location of nodes. Main problems are the occurrence of thehalf-integer powers in the set of ansatz functions and the fact that the coefficientsak and ak are not independent.

Now we formulate the following theorem:

Theorem 3.1. For n given arbitrary nodes ϕ0, ϕ1, . . . , ϕn−1 basis functions of theform (3.1) exist, satisfying the canonical interpolation problem

f 〈i〉n (ϕk) = δ(i−1)k, k = 0, . . . , n− 1, (3.2)

where i = 1, . . . , n is the number of a canonical problem.

Proof. Without loss of generality we will consider here the first canonical problem

for f〈1〉n . In all upcoming calculations we take rA = 1. We start our proof by

introducing the new variable

t = eiϕ2 , |t| = 1.

The function (3.1) can then be rewritten as

fn(t) =

N1∑k=0,2,...

[akκ t

k + akκ t−k +

k

2ak t

−k − k

2ak t

−k+4

]

+

N2∑k=1,3,...

[akκ t

k − akκ t−k +

k

2ak t

−k − k

2ak t

−k+4

].

(3.3)

Depending on the number n of nodes on the interface ΓAD we have a differentnumber of functions from the even and odd parts of the basis. For the case of aneven number of nodes we have N2 = N1 + 1 and in the case of an odd number


of nodes we have N1 = N2 + 1. This fact must be taken into account during theproof.

Let us consider at first the case when the number of nodes n is even. In thiscase we can write the interpolation function (3.3) as one finite sum

fn(t) =

12n−1∑k=0

[2k + 1

2a2k+1 t

−2k−1 + a2kκ t−2k + ka2k t

−2k

− ka2k t−2k+4 + a2k+1κ t

2k+1 − a2k+1κ t−2k−1

+a2kκ t2k − 2k + 1

2a2k+1 t

−2k+3

].

(3.4)

The interpolation problem (3.2) reads as follows:

fn(tj) = δ0j , j = 0, . . . , n− 1.

Analysing equation (3.4) we observe that the lowest degree is −n+1. There-fore, to obtain a polynomial we multiply both sides of the (3.4) by tn−1 and weget at the nodes

12n−1∑k=0

[2k + 1

2a2k+1 t

n−2k−2j + a2kκ t

n−2k−1j + ka2k t

n−2k−1j

− ka2k tn−2k+3j + a2k+1κ t

n+2kj − a2k+1κ t

n−2k−2j

+a2kκ tn+2k−1j − 2k + 1

2a2k+1 t

n−2k+2j

]= δ0j t

n−1j .

Now, we introduce a new right-hand side

wj := δ0j tn−1j , j = 0, . . . , n− 1.

Collecting all summands with the same degree we can write the polynomial withnew coefficients αl

2n−2∑l=0

αltl :=

12n−1∑k=0

[2k + 1

2a2k+1 t

n−2k−2 + a2kκ tn−2k−1

+ ka2k tn−2k−1 − ka2k t

n−2k+3

+ a2k+1κ tn+2k − a2k+1κ t

n−2k−2

+a2kκ tn+2k−1 − 2k + 1

2a2k+1 t

n−2k+2

].

(3.5)

Thus we get the following equivalent interpolation problem

2n−2∑l=0

αltlj = wj , j = 0, . . . , n− 1. (3.6)

Due to the fact that the polynomial (3.5) contains also shifted powers tn−2k+3 andtn−2k+2, the equations relating the new coefficients αl with the original coefficients


ak will change the form with increasing number of nodes. We will consider hereonly the case n > 6, because then these relating equations take their general formand the proof applies for all n > 6. The remaining three cases n = 2, n = 4,and n = 6 can be easily obtained directly from (3.6) and will not influence thegenerality of the proof.

For n > 6 we can separate the following four groups of equations between αl

and ak

(I)

{α2j =

n−1−2j2 an−1−2j − κ an−1−2j,

α2j+1 = κ an−2−2j +(n2 − 1− j

)an−2−2j ,

j = 0, 1

(II)

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩α2j =

n−1−2j2 an−1−2j − κ an−1−2j − n+3−2j

2 an+3−2j ,

j = 2, . . . , n−22

α2j+1 = κ an−2−2j +(n2 − 1− j

)an−2−2j −

(n2 + 1− j

)an+2−2j ,

j = 2, . . . , n−42

(III)

{α2j = κ a2j−n+1 +

2j−n−32 an+3−2j,

α2j−1 =(n2 − j + 2

)κ a2j−n −

(n2 − j + 2

)a2j−n,

j = n2 ,

n+22

(IV)

{α2j = κ a2j+1−n,

α2j−1 = κ a2j−nj = n+4

2 , . . . , n− 1. (3.7)

From the equations (3.7) we can calculate explicitly all of the original coefficientsal. The group (IV) leads to the following equations:{

a2j−n =α2j−1

κ ,

a2j+1−n =α2j

κ ,j = n

2 + 2, . . . , n− 1. (3.8)

This group of equations includes all of the coefficients al for l = 4, . . . , n − 1.Therefore we need only to add equations for the four remaining coefficients. Wecan calculate the coefficients a0, a1, a3 from group (III) of equations (3.7). Thecoefficient a2 can be obtained from the sum of the third equation in group (III)and equation n− 6 from group (II). These coefficients are given by⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

a0 = αn−1

2κ + αn+3

κ2 ,

a1 =[αn

κ + 3αn+2

2κ2

] (1− 3

4κ2

)−1,

a2 = αn−3+αn+1

2κ + 3αn+5

2κ2 ,

a3 = αn+2

κ + 12κ

[αn

κ + 3αn+2

2κ2

] (1− 3

4κ2

)−1.

(3.9)

The interpolation problem (3.6) contains 2n− 1 unknown coefficients αl, butfrom the interpolation nodes we can get only n equations. Therefore, we formulaten − 1 additional equations to determine all coefficients αl. For that reason weextend (3.5) to the whole complex plane and add n−1 Hermite-type interpolation


conditions2n−2∑l=0,1

αl

(∂ptl

∂tp

)t=t∗

= w∗j , j = 0, . . . , 2n− 2, (3.10)

where for simplicity we take the additional node t∗ at 0 and the values w∗j are

defined as follows:

w∗j =

{0, j = 0, . . . , n− 5,βj , j = n− 4, . . . , n− 2,

(3.11)

with arbitrary complex numbers βj .The obtained “extended” interpolation problem (3.2), (3.10) is always solv-

able (for more details, see for instance [5]). The solution of this interpolationproblem will give us all coefficients αl, which are needed to define the originalcoefficients ak.

To insure that the coefficients ak satisfy the original interpolation prob-lem (3.2) we need to satisfy compatibility conditions for the coefficients αk. Byusing formulae (3.8)–(3.9) we obtain these conditions from the groups (I) and (II)of equations (3.7). The first group gives the following four equations

α2j + α2n−2−2j − (n− 2j − 1)α2n−2−2j

2κ= 0,

α2j+1 − α2n−3−2j −(n− 2

2− j

)α2n−3−2j

κ= 0,

(3.12)

for j = 0, 1. The second group leads to the remaining n− 5 equations

α2j + α2n−2−2j − (n− 2j − 1)α2n−2−2j

2κ+ (n− 2j + 3)

α2n−2(j−1)

2κ= 0,

α2j+1 − α2n−3−2j −(n− 2

2− j

)α2n−3−2j

κ+

(n− 2

2+ 2− j

)α2n−2j+1

κ= 0,

(3.13)for j = 2, . . . , n−6

2 .

αn+2 +1

2

[αn

κ+

3αn+2

2κ2

](1− 3

4κ2

)−1

+ αn−4 +7αn+6

2κ

− 3αn+2

2κ− 3

4κ

[αn

κ+

3αn+2

2κ2

](1− 3

4κ2

)−1

= 0,

αn−3 − αn−3 + αn+1

2− 3αn+5

2κ− αn−3 + αn+1

2κ− 3αn+5

2κ2+

3αn+5

κ= 0,

(3.14)

κ

[αn

κ+

3αn+2

2κ2

](1− 3

4κ2

)−1

+ αn−2 +5αn+4

2κ

− 1

2

[αn

κ+

3αn+2

2κ2

](1− 3

4κ2

)−1

= 0.

(3.15)

Our goal is to show that there exists a set of complex numbers βj such thatthe compatibility conditions (3.12)–(3.15) are satisfied. Considering the first n− 4


equations of (3.10) with the values (3.11) we get immediately that

αl = 0, i = 0, . . . , n− 5. (3.16)

Applying these results to first n − 4 compatibility conditions (3.12)–(3.13) weobtain

αl = 0, i = n+ 3, . . . , 2n− 2. (3.17)

Therefore first n− 4 compatibility conditions are satisfied, and we need to checkthe remaining three equations (3.14)–(3.13). Taking into account the remainingthree values of w∗

j we get the following equations

αn+2 +1

2

[αn

κ+

3αn+2

2κ2

](1− 3

4κ2

)−1

+ βn−4 − 3αn+2

2κ

− 3

4κ

[αn

κ+

3αn+2

2κ2

](1− 3

4κ2

)−1

= 0,

βn−3 − βn−3 + αn+1

2− βn−3 + αn+1

2κ= 0,

κ

[αn

κ+

3αn+2

2κ2

](1− 3

4κ2

)−1

+ βn−2 − 1

2

[αn

κ+

3αn+2

2κ2

](1− 3

4κ2

)−1

= 0.

The second equation can be satisfied only if αn+1 = βn−3 = 0. The solution of thetwo other equations is given by

αn = − 4κ2 �[βn−2](4κ2−3)

(1− 1

2κ

) +6κ�[βn−4]

(4κ2−3)(1− 3

2κ

) − 4i κ2 �[βn−2](4κ2−3)

(1+

12κ

) − 6i κ�[βn−4](4κ2−3)

(1+

32κ

) ,

αn+2 = − 4κ2 �[βn−4](4κ2−3)

(1− 3

2κ

) +2κ�[βn−2]

(4κ2−3)(1− 1

2κ

) − 4i κ2 �[βn−4](4κ2−3)

(1+

32κ

) − 2i κ�[βn−2](4κ2−3)

(1+

12κ

) ,

where

βn−2 := βn−2

(1− 3

4κ2

), βn−4 := βn−4

(1− 3

4κ2

).

Finally, we obtained that all compatibility conditions are satisfied. Thus we haveshown, that such set of complex numbers βj exists, and the statement of thetheorem is true for the case of even number of nodes.

Now we will complete the proof by considering the case when the number ofnodes n is odd. We will omit some details which are similar to the even case. In thecase of an odd number of nodes we need to keep the structure of the interpolationfunction (3.3) with two separate sums

fn(t) =

n−12∑

k=0,1

[a2kκ t

2k + a2kκ t−2k + ka2k t

−2k − ka2k t−2k+4

]

+

n−12∑

k=1,2

[a2k−1κ t

2k−1 − akκ t−2k+1 + 2k−1

2 a2k−1 t−2k+1 − 2k−1

2 a2k−1 t−2k+5

].


To simplify the above function we extract the term for k = 0 from the first sum,collect common terms and get

fn(t) = 2a0κ+

n−12∑

k=1,2

[t−2k(a2kκ+ k a2k)

+ t−2k+1

(2k − 1

2a2k−1 − a2k−1κ

)− k a2kt

−2k+4

− 2k − 1

2a2k−1t

−2k+5 + a2k−1κ t2k−1 + a2kκ t

2k

].

(3.18)

The lowest degree is −n+ 1. Therefore, as in the previous case we multiplyboth sides of the interpolation problem by tn−1, and we get the following equivalentinterpolation problem

2n−2∑l=0

αltlj = wj ,

for j = 0, . . . , n− 1.Since the polynomial basis contains also shifted powers tn−2k+3 and tn−2k+4

the equations relating the new coefficients αl with the original coefficients ak willchange with increasing n and take their general form for n > 7. We will consideronly this case. The remaining three cases n = 3, n = 5, and n = 7 can beeasily obtained directly from the interpolation problem and will not influence thegenerality of the proof.

Similar to the even case, for n > 7 we can get four groups of equationsbetween αk and ak

(I)

{α2j = κ an−1−2j +

(n−12 − j

)an−1−2j ,

α2j+1 = n−2−2j2 an−2−2j − κ an−2−2j,

j = 0, 1

(II)

⎧⎪⎨⎪⎩α2j = κ an−1−2j +

(n−12 − j

)an−1−2j −

(n+32 − j

)an+3−2j,

α2j+1 = n−2−2j2 an−2−2j − κ an−2−2j − n+2−2j

2 an+2−2j ,

j = 2, . . . , n−32

(III)

{α2j =

(n−12 − j + 2

)κ a2j−n+1 − n+3−2j

2 an+3−2j,

α2j−1 = κ a2j−n+2 +2j−n−2

2 an+2−2j ,j = n−1

2 , n+12

(IV)

{α2j = κ a2j−n+1,

α2j−1 = κ a2j−n+2,j = n+3

2 , . . . , n− 1. (3.19)

Analogously to the even case, from equations (3.19) we get the explicit for-mulae for the coefficients ak. From group (IV) we get the following equations forak for k = 4, . . . , n− 1{

a2j−n+1 =α2j

κ ,

a2j−n+2 =α2j−1

κ ,j = n+3

2 , . . . , n− 1. (3.20)


Formulae for the remaining coefficients a0, a1, a2, a3 are completely the same as inthe even case, and they are given by (3.9).

Applying the same ideas as in the case of even number of nodes we introducesome additional Hermite-type conditions (3.10). The remaining task is to provethe compatibility conditions for the case of an odd number of nodes. From the firstgroup we get the following equations{

α2j − α2n−2−2j −(n−12 − j

) α2n−2−2j

κ = 0,

α2j+1 + α2n−3−2j − (n− 2j − 2)α2n−3−2j

2κ = 0,j = 0, 1. (3.21)

The second group leads to the following n− 5 equations:⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

α2j − α2n−2j−2 −(n−12 − j

) α2n−2−2j

κ

+(n−12 − j + 2

) α2n−2(j−1)

κ = 0, j = 2, . . . , n−52

α2j+1 − α2n−2j−3 − (n− 2j − 2)α2n−3−2j

2κ

+ (n− 2j + 2)α2n−2j+1

2κ = 0, j = 2, . . . , n−72

αn+2 +12

[αn

κ + 3αn+2

2κ2

] (1− 3

4κ2

)−1+ αn−4 +

7αn+6

2κ

− 3αn+2

2κ − 34κ

[αn

κ + 3αn+2

2κ2

] (1− 3

4κ2

)−1= 0,

αn−3 − αn−3+αn+1

2 − 3αn+5

2κ − αn−3+αn+1

2κ − 3αn+5

2κ2 + 3αn+5

κ = 0,

κ[αn

κ + 3αn+2

2κ2

] (1− 3

4κ2

)−1+ αn−2 +

5αn+4

2κ

− 12

[αn

κ + 3αn+2

2κ2

] (1− 3

4κ2

)−1= 0.

(3.22)

In a similar way as for the case of an even number of nodes it can be shown,that there exists a set of complex numbers βj such that the compatibility conditionsare satisfied and the statement of the theorem is true for the case of an odd numberof nodes. �

4. Conclusions and outlook

It has been shown that the canonical interpolation problems (3.2) can be solvedfor an arbitrary number of nodes. Due to complexity of the basis function it wasnecessary to consider separately the cases for even and odd number of interpolationnodes. The obtained result is very important for the convergence analysis of theproposed method. Particularly, the number on nodes on the interface and thenumber of the coupling elements can have a significant influence on the errorestimate of the proposed scheme.

Acknowledgement

The research of the second author is supported by the German Research Foun-dation (DFG) via Research Training Group “Evaluation of Coupled Numerical


Partial Models in Structural Engineering (GRK 1462)”, which is gratefully ac-knowledged.

References

[1] Philippe G. Ciarlet, The finite element method for elliptic problems, North-HollandPublishing Company, 1978.

[2] S. Bock, K. Gurlebeck, D. Legatiuk, Convergence of the finite element method withholomorphic functions. AIP Conference proceedings 1558, 513 (2013).

[3] S. Bock, K. Gurlebeck, D. Legatiuk, On a special finite element based on holomorphicfunctions, AIP Conference proceedings 1479, 308 (2012).

[4] S. Bock, K. Gurlebeck, D. Legatiuk, On the continuous coupling between analyticaland finite element solutions, Le Hung Son & Wolfgang Tutschke, eds. Interactionsbetween real and complex analysis, pp. 3–19. Science and Technics Publishing House,Hanoi, 2012.

[5] Philip J. Davis Interpolation and Approximation, Dover Publications, Inc., 1975.

[6] D. Legatiuk, K. Gurlebeck, G. Morgenthal, Modelling of concrete hinges throughcoupling of analytical and finite element solutions. Bautechnik Sonderdruck, ISSN0932-8351, A 1556, April 2013.

[7] H. Liebowitz, Fracture, an advanced treatise. Volume II: Mathematical fundamentals,Academic Press, 1968.

[8] N.I. Mußchelischwili, Einige Grundaufgaben der mathematischen Elastizitatstheorie,VEB Fachbuchverlag Leipzig, 1971.

[9] R. Piltner, Some remarks on finite elements with an elliptic hole, Finite elements inanalysis and design, Volume 44, Issues 12-13, 2008.

[10] R. Piltner, Special finite elements with holes and internal cracks, International jour-nal for numerical methods in engineering, Volume 21, 1985.

[11] A. Schumann, Untersuchung und Beurteilung des Rissverhaltens eines Betongelenkesanhand unterschiedlicher Methoden, Bachelorarbeit Nr. BB/2013/8. Supervised byK. Gurlebeck and D. Legatiuk.

Klaus GurlebeckChair of Applied MathematicsBauhaus-University WeimarD-99423 Weimar, Germanye-mail: [email protected]

Dmitrii LegatiukResearch Training Group 1462Bauhaus-University WeimarD-99425 Weimar, Germanye-mail: [email protected]





On ψ-hyperholomorphic Functionsand a Decomposition of Harmonics

Klaus Gurlebeck and Hung Manh Nguyen

Abstract. Additive decompositions of harmonic functions play an importantrole in function theory and for the solution of partial differential equations.One of the best known results is the decomposition of harmonic functions as asum of a holomorphic and an anti-holomorphic function. This decompositioncan be generalized also to the analysis of quaternion-valued harmonic function,where the summands are then monogenic or anti-monogenic, respectively.For paravector-valued functions, sometimes called A-valued functions, thisdecomposition is not possible. The main purpose of the paper is to showthat one can find three different ψ-Cauchy–Riemann operators such that thenull spaces of these operators define an additive decomposition for harmonicfunctions, mapping R3 to A=R3.

Mathematics Subject Classification (2010). 30G35, 42C05, 33E10.

Keywords. Quaternion analysis, ψ-hyperholomorphic functions.

1. Introduction

The theory of monogenic functions as theory of the null solutions of a Dirac op-erator or a generalized Cauchy–Riemann operator can be seen as a refinementof harmonic analysis or as a generalization of complex analysis. One of the mostimportant properties of monogenic (or holomorphic) functions is that they areharmonic functions in all components of the vector functions. Already in 1989 inthe thesis by Stern [17] (see also [18]) the question was asked which properties ofa first-order partial differential operator ensure that all null solutions of this oper-ator are harmonic in all components. It was shown that the coefficients (matricesin this work) must satisfy the multiplication rules of a Clifford algebra.

Independent on this research Shapiro and Vasilevski introduced in the late1980’s the theory of so-called ψ-hyperholomorphic quaternion-valued functions

The second author acknowledges the financial support of MOET-Vietnam & DAAD.

Switzerland

182 K. Gurlebeck and H. Manh Nguyen

(see [14] and later [15]). In this theory the standard basis vectors from quater-nionic analysis are replaced by a more general structural set. Seen as vectors fromR4 the elements of the structural set must be an orthonormal set with respectto the standard inner product in R4. The authors used this approach to studysome singular integral operators in spaces of quaternion-valued functions. In par-ticular a generalized Π-operator was studied and relations of this Π-operator tothe Bergmann projection could be proved. The work on the Π-operator continuedsome earlier work by Shevchenko [16] who studied special Π operators based onmodified generalized Cauchy–Riemann operators which are covered by the theoryof ψ-hyperholomorphic functions.

In 1998 it was shown in [5] that the class of ψ-hyperholomorphic functions ismore than what we get by rotations from the class of monogenic functions. In thisline are also the results in [7] where it could be shown that a special Π operator isinvertible in L2 and how the mapping properties of the operator change with thestructural set. Recently this topic was studied again in [1] for Π operators definedon domains with fractal boundaries.

A second line of the research on ψ-hyperholomorphic functions is relatedto their geometric mapping properties. In [8] and [9] it was observed that ψ-hyperholomorphic functions can be connected with certain conformal mappings.Later on Malonek introduced in [11] the concept of M-conformal mappings that isalso related to ψ-hyperholomorphic mappings

The third line of research is concerned with the refinement of harmonic anal-ysis. This refinement is based on the factorization of the Laplacian by Dirac op-erators or by generalized Cauchy–Riemann operators. One of the basics of thetheory of ψ-hyperholomorphic functions is that the structural sets must be chosenin a way that this factorization also holds for ψ-Cauchy–Riemann operators. Asecond question is to find additive decompositions of harmonic functions. It is wellknown in complex analysis that a harmonic function can be decomposed as a sumof a holomorphic and an anti-holomorphic function. An analogous result holds forH-valued harmonic functions which can be represented as a sum of a monogenicand an anti-monogenic H-valued function.

Recently the theory of A-valued monogenic and harmonic functions foundsome interest. Motivated by applications in R3 and the observation that A-valuedfunctions share more properties with holomorphic functions [12, 13] than generalH-valued monogenic functions the question of additive decompositions was studiedagain for harmonic functions in R3. Alvarez and Porter [2] made the surprisingobservation that A-valued functions cannot be written as a sum of a monogenicand an anti-monogenic A-valued function.

They found that in the 6n+3-dimensional subspace of homogeneous harmonicpolynomials of degree n there is a 2n− 1-dimensional subspace orthogonal to thesum of monogenic and anti-monogenic polynomials of the same degree, calledcontragenic functions.

It will be shown in the paper that contragenic functions cannot be solutions ofa first-order system of partial differential equations. So, the main question is if there

On ψ-hyperholomorphic Functions 183

are other first-order systems such that we can decompose harmonic functions as asum of three subspaces of null solutions of first-order systems of partial differentialequations with the property that all solutions of those systems are harmonic in allcoordinates.

To answer the fundamental question of the existence of such additive decom-positions we will study ψ-hyperholomorphic functions, ψ-anti-holomorphic func-tions and construct another structural set θ such that the corresponding null spacesgive us the desired decomposition. We will not consider the problem to find allpossible decomposition here.

2. Preliminaries

Let H be the algebra of real quaternions generated by the basis {1, e1, e2, e3}subject to the multiplication rules

eiej + ejei = −2δij, i, j = 1, 2, 3

e1e2 = e3.

Each quaternion can be represented in the form q = q0+ q1e1+ q2e2+ q3e3 whereqj (j = 0, . . . , 3) are real numbers. The real and vector parts of q are denoted bySc (q) := q0 and Vec (q) := q1e1 + q2e2 + q3e3. The real vector space R3 will beembedded in H by identifying the element x = (x0, x1, x2) ∈ R3 with the reducedquaternion x = x0 + x1e1 + x2e2. The set of all reduced quaternions is denotedby A which is a R-linear subspace of H. Let B be the unit ball in R3. L2(B,A) iscalled the right R-linear Hilbert space of all square integrable A-valued functionsin B, endowed with the real-valued inner product

〈f, g〉 =∫B

Sc (fg) dω (2.1)

where dω is the Lebesgue measure in R3. The generalized Cauchy–Riemann oper-ator and its adjoint operator are given by

∂ :=∂

∂x0+ e1

∂

∂x1+ e2

∂

∂x2,

∂ :=∂

∂x0− e1

∂

∂x1− e2

∂

∂x2.

Definition 2.1. A function f ∈ C1(B,A) is called (left-) monogenic in B ⊂ R3 if∂f = 0 in B.

Definition 2.2. Let f ∈ C1(B;A) be a continuous, real differentiable function andmonogenic in B. The expression

(12∂f

)is called hypercomplex derivative of f in B.

To be short, we simply present the hypercomplex derivative as a definition.In the paper [6], it is proved that monogenicity and hypercomplex derivabilityare equivalent in all dimensions, and that the hypercomplex linearization of themonogenic function f is exactly given by 1

2∂f . A complete survey on this topiccan be found also in [10].


3. Quaternionic ψ-hyperholomorphic functions

The definition of quaternion-valued ψ-hyperholomorphic functions was studiedby M.V. Shapiro and N.L. Vasilevski in [15] as a generalization of monogenicfunctions with respect to the basis {1, e1, e2, e3}. One can also find researches onψ-hyperholomorphic functions by R. Delanghe, R.S. Kraußhar and H.R. Malonek[9]; R.S. Kraußhar and H.R. Malonek [8]. Following the same ideas, we considerthe case in R3.

Let ψ := {ψ0, ψ1, ψ2} ⊂ A and ψ := {ψ0, ψ1, ψ2}. The generalized Cauchy–Riemann operator ψD is defined by

ψD[f ] := ψ0∂0f + ψ1∂1f + ψ2∂2f.

To fulfil the Laplacian factorization ΔR3 = ψDψD = ψDψD, the following condi-tion holds

ψjψk + ψkψj = 2δjk (3.1)

for j, k = 0, . . . , 2. A set ψ satisfying the relation (3.1) will be called a structuralset. This can be interpreted geometrically. Suppose that we have a structural

set ψ = {ψ0, ψ1, ψ2} ⊂ A which is identified with a vector set−→ψ in R3. As a

consequence, the vector set−→ψ forms an orthonormal basis in R3. Particularly, if

we write

ψ0 = ψ00 + ψ0

1e1 + ψ02e2

ψ1 = ψ10 + ψ1

1e1 + ψ12e2

ψ2 = ψ20 + ψ2

1e1 + ψ22e2

then one gets a formal matrix representation

(ψ0 ψ1 ψ2

)=(1 e1 e2

)⎛⎜⎜⎝ψ00 ψ1

0 ψ20

ψ01 ψ1

1 ψ21

ψ02 ψ1

2 ψ22

⎞⎟⎟⎠︸︷︷︸

Ψ

.

By virtue of (3.1), the matrix Ψ is an orthogonal matrix, i.e., ΨΨ′ = Ψ′Ψ = I

(where I is the 3× 3- unit matrix). Correspondingly, {−→ψ } is an orthonormal basisof R3.

A C1(B,A) function f is called an (A-valued) ψ-hyperholomorphic functionin B if it satisfies ψDf(x) = 0 for all x ∈ B. The monogenic case corresponds withthe standard structural set {1, e1, e2}. We refer readers to [15] for a survey on ψ-hyperholomorphic functions, the ψ-hypercomplex derivative and Cauchy integrals.To this end, ψM(B,A) stands for the L2-space of ψ-hyperholomorphic functionsin B and ψM(B,A, n) is its subspace of homogeneous ψ-hyperholomorphic poly-nomials of degree n. For the standard structural set, we use the notationM only.The notationM means the space of conjugations of functions in M. In fact, thatis the space of anti-monogenic functions shown in [3].


4. Contragenic functions revisited

It is well known that a Cln,0-valued harmonic function can be decomposed into thesum of a (Cln,0-valued) monogenic and an anti-monogenic function, see [4]. How-ever this is not the case for A-valued harmonic functions. Particularly, due to [2],H(B,A, n) stands for the space of A-valued homogeneous harmonic polynomialsof degree n in B, then with n > 0

dimH(B,A, n) = 6n+ 3

while

dim(M(B,A, n) +M(B,A, n)) = 4n+ 4.

It shows that there are harmonic functions which can not be the sum of a mono-genic and an anti-monogenic function. The orthogonal complement in the spaceH(B,A, n), denoted by

N (B,A, n) := (M(B,A, n) +M(B,A, n))⊥is called the space of homogeneous contragenic polynomials of degree n. It yields

dimN (B,A, n) = 2n− 1.

Having in mind that the following 2n + 1 spherical harmonic functions form awell-known orthogonal basis of the space of real-valued homogeneous harmonicpolynomials of degree n ≥ 0

Un0 , U

n1 , . . . , U

nn , V

n1 , . . . , V n

n

where Unm, V n

l are defined in spherical coordinates x0 = r cos θ, x1 = r sin θ cosϕ,x2 = r sin θ sinϕ, via

Unm = rnPm

n (cos θ) cos(mϕ), m = 0, . . . , n

V nl = rnP l

n(cos θ) sin(lϕ), l = 1, . . . , n.

Pmn is the associated Legendre function

Pmn (x) = (−1)m(1− x2)m/2(dm/dxm)Pn(x)

with Pn is the Legendre polynomial of degree n corresponding to m = 0. Ac-cording to [2, 12], we have explicit orthogonal bases of the spacesM(B,A, n) andN (B,A, n).Proposition 4.1. Denote by cnm = (n+m)(n+m+ 1)/4. For each n ≥ 1, thefollowing functions form an orthogonal basis of M(B,A, n)

Xn0 =

n+ 1

2Un0 +

1

2Un1 e1 +

1

2V n1 e2

Xnm =

n+m+ 1

2Unm +

(14Unm+1 − cnmUn

m−1

)e1 +

(14V nm+1 + cnmV n

m−1

)e2

Y nm =

n+m+ 1

2V nm +

(14V nm+1 − cnmV n

m−1

)e1 −

(14Unm+1 + cnmUn

m−1

)e2,

where 1 ≤ m ≤ n+ 1.


Proposition 4.2. Let n ≥ 1 and dnm = (n −m)(n −m + 1). The following 2n − 1functions

Zn0 = V n

1 e1 − Un1 e2

Znm,+ =

(dnmV n

m−1 + V nm+1

)e1 +

(dnmUn

m−1 − Unm+1

)e2

Znm,− =

(dnmUn

m−1 + Unm+1

)e1 +

(−dnmV n

m−1 + V nm+1

)e2

for 1 ≤ m ≤ n− 1, form an orthogonal basis of N (B,A, n).Let the ambigenic function space be the sum of monogenic and anti-mono-

genic spaces. Finally, an A-valued harmonic function can be orthogonally decom-posed into an ambigenic and a contragenic function. Details can be found in [2].

The contragenic function space is defined formally as the orthogonal comple-ment of the ambigenic function space. Of course, contragenic functions are alsoharmonic functions. It means that they satisfy the Laplace equation as well asmonogenic functions. We know that monogenic functions are null solutions of the(generalized) Cauchy–Riemann operator. The question is which first-order linearpartial differential operator characterizes contragenic functions. Particularly, itwould be useful if we can find a structural set ψ such that contragenic functionsare solutions of the operator ψD. The answer will lead to another decompositionfor harmonic functions.

5. An additive decomposition of A-valued harmonic functions

In this section, we will prove that an A-valued harmonic function can be decom-posed into the sum of three different ψ-hyperholomorphic functions. Firstly, onecan see that there does not exist any structural set ψ such that contragenic func-tions are null-solutions of the corresponding Cauchy–Riemann operator. Indeed,consider a Cauchy–Riemann operator

ψD = ψ0 ∂

∂x0+ ψ1 ∂

∂x1+ ψ2 ∂

∂x2.

Apply ψD to two first contragenic basis functions in [2]

Z10 = −x2e1 + x1e2

Z20 = 3x0(−x2e1 + x1e2)

and let the results be zero, one gets ψ0 = 0. This contradicts the definition ofψ. Moreover, one can prove that contragenic functions are not solutions of anyfirst-order linear partial differential operator.

Now, we are looking for a structural set ψ such that the sum of the spacesof (corresponding) ψ-hyperholomorphic, monogenic and anti-monogenic functionsis the (A-valued) harmonic function space. It can be proved that if f := f0 +


f1e1 + f2e2 is a monogenic function then ψf := ψ0f0 − ψ1f1 − ψ2f2 is a ψ-hyperholomorphic function, i.e., ψDψf = 0. Therefore, we have the followinglemma.

Lemma 5.1. Let ψ = {1, e2, −e1}. The following functions form an orthogonalbasis for ψM(B,A, n)

ψXn0 =

n+ 1

2Un0 +

1

2Un1 e2 −

1

2V n1 e1

ψXnm =

n+m+ 1

2Unm +

(14Unm+1 − cnmUn

m−1

)e2 −

(14V nm+1 + cnmV n

m−1

)e1

ψY nm =

n+m+ 1

2V nm +

(14V nm+1 − cnmV n

m−1

)e2 +

(14Unm+1 + cnmUn

m−1

)e1

with 1 ≤ m ≤ n+ 1.

Remark that with ψ = {1, e2, −e1}, we have

ψXnn+1 = Y n

n+1ψY n

n+1 = −Xnn+1

ψXnn =

1

2

(Xn

n +Xnn

)+

1

2

(Y nn − Y n

n

)ψY n

n =1

2

(−Xnn +Xn

n

)+

1

2

(Y nn + Y n

n

).

Next, we prove an additive decomposition of A-valued harmonic functions.

Theorem 5.2. Let ψ = {1, e2, −e1}. Every A-valued harmonic function u can bedecomposed into the form

u = f + f1 + f2

where f , f1, f2 are A-valued monogenic, anti-monogenic and ψ-hyperholomorphicfunctions, respectively.

Proof. Remind that an A-valued harmonic function is the sum of monogenic, anti-monogenic and contragenic functions (see [2]). Therefore, in order to prove thistheorem, we will show that each contragenic function can be linearly representedby monogenic, anti-monogenic and ψ-hyperholomorphic functions. This in turn isrestricted in the case of polynomials of degree n ≥ 1. Indeed, for m = 0 we have

Zn0 = −2 ψXn

0 +Xn0 +Xn

0 .

The contragenic polynomial Znm,+, 1 ≤ m ≤ n− 1, can be rewritten as follows:

Znm,+ = V n

m+1e1 − Unm+1e2 + dnm

(V nm−1e1 + Un

m−1e2

).

This suggests that Znm,+ can be described in the form

Znm,+ = αn

m,+

{ψXn

m −1

2

(Xn

m +Xnm

)− βnm,+

(Y nm − Y n

m

)}.

Straightforward calculations lead to the system{αnm,+

(− 14 − 1

2βnm,+

)= 1

αnm,+

(−cnm + 2cnmβnm,+

)= dnm.


This system has the solution⎧⎨⎩αnm,+ = − 4cnm+dn

m

2cnm= − 4(n2+m2+n)

(n+m)(n+m+1)

βnm,+ =

4cnm−dnm

2(4cnm+dnm) =

m(2n+1)2(n2+m2+n) .

Analogously, we can also find the representation of Znm,− in terms of Xn

m, Xnm, Y n

m,

Y nm and ψY n

m

Znm,− = αn

m,−

{ψY n

m − βnm,−

(Xn

m −Xnm

)− 1

2

(Y nm + Y n

m

)}where ⎧⎨⎩αn

m,− =dnm+4cnm2cnm

= 4(n2+m2+n)(n+m)(n+m+1)

βnm,− =

dnm−4cnm

2(dnm+4cnm) = − m(2n+1)

2(n2+m2+n) .

This completes our proof. �

Different from the decomposition by means of contragenic functions, thisdecomposition is not orthogonal. However, every component in the decompositionshares the same structure as monogenic functions. The advantage is that nowin each subspace of the decomposition all tools from quaternionic analysis likeintegral representations and kernel functions are available.

6. Conclusion

The theory of ψ-hyperholomorphic functions shows a structural analogy with clas-sical monogenic functions. It helps to have a better understanding about character-istics of monogenic and harmonic functions such as geometric mapping propertiesand harmonic decompositions. The question how to find all possible harmonicdecompositions is still open.

References

[1] R. Abreu Blaya, J. Bory Reyes, A. Guzman Adan, and U. Kaehler, On some struc-tural sets and a quaternionic (φ,ψ)-hyperholomorphic function theory, submitted toMathematische Nachrichten.

[2] C. Alvarez-Pena and R. Michael Porter, Contragenic Functions of Three Variables,Complex Anal. Oper. Theory, 8 (2014), 409–427.

[3] S. Bock, On a three-dimensional analogue to the holomorphic z-powers: power seriesand recurrence formulae, Complex Variables and Elliptic Equations: An Interna-tional Journal, 57 (2012), 1349–1370.

[4] F. Brackx, R. Delanghe and F. Sommen, Clifford Analysis, Pitman Publishing,Boston-London-Melbourne, 1982.

[5] K. Gurlebeck: On some classes of Pi-operators, in Dirac operators in analysis, (eds.J. Ryan and D. Struppa), Pitman Research Notes in Mathematics, No. 394, 1998.


[6] K. Gurlebeck and H. Malonek, A hypercomplex derivative of monogenic functions inRn+1 and its applications, Complex Variables, 39, No. 3 (1999), 199–228.

[7] K. Gurlebeck, U. Kahler, M. Shapiro: On the Pi-operator in hyperholomorphic func-tion theory, Advances in Applied Clifford Algebras, Vol. 9(1), 1999, pp. 23–40

[8] R.S. Kraußhar, H.R. Malonek, A characterization of conformal mappings in R4 bya formal differentiability condition. Bull. Soc. R. Sci. Liege 70, No. 1, 35–49 (2001).

[9] R. Delanghe, R.S. Kraußhar, H.R. Malonek, Differentiability of functions with valuesin some real associative algebras: approaches to an old problem. Bull. Soc. R. Sci.Liege 70, No. 4-6, 231–249 (2001).

[10] M.E. Luna Elizarraras and M. Shapiro, A survey on the (hyper)derivates in complex,quaternionic and Clifford analysis, Millan J. of Math, 79 (2011), 521–542.

[11] H.R. Malonek, Contributions to a geometric function theory in higher dimensionsby Clifford analysis method: Monogenic functions and M-conformal mappings. InBrackx, F. (ed.) et al., Clifford analysis and its applications. Proceedings of theNATO advanced research workshop, Prague, Czech Republic, October 30–November3, 2000. Dordrecht: Kluwer Academic Publishers. NATO Sci. Ser. II, Math. Phys.Chem. 25, 213–222 (2001).

[12] J. Morais and K. Gurlebeck, Real-Part Estimates for Solutions of the Riesz Systemin R3, Complex Var. Elliptic Equ., 57 (2012), 505–522.

[13] J. Morais, K. Gurlebeck: Bloch’s Theorem in the Context of Quaternion Analysis,Computational Methods and Function Theory, Vol. 12 (2012), No. 2, 541–558.

[14] Vasilevskij, N.L.; Shapiro, M.V., On Bergmann kernel functions in quaternion anal-ysis. Russ. Math. 42, No. 2, 81–85 (1998); translation from Izv. Vyssh. Uchebn.Zaved., Mat. 1998, No. 2, 84–88 (1998).

[15] M.V. Shapiro and N.L. Vasilevski, Quaternionic ψ-hyperholomorphic functions, sin-gular integral operators and boundary value problems. I. ψ-hyperholomorphic functiontheory, Complex Variables, 27 (1995), 17–46.

[16] V.I. Shevchenko, A local homeomorphism of 3-space realizable by the solution of acertain elliptic system, Dokl. Acad. Nauk 146 (1962), 1035–1038.

[17] I. Stern, Randwertaufgaben fur verallgemeinerte Cauchy–Riemann-Systeme imRaum. Dissertation A, Martin-Luther-Universitat Halle-Wittenberg 1989.

[18] I. Stern, Boundary value problems for generalized Cauchy–Riemann systems in thespace. In: Kuhnau R.; Tutschke, W. (eds.): Boundary value and initial value problemsin complex analysis. Pitman Res. Notes Math. 256 (1991): 159–183.

Klaus Gurlebeck and Hung Manh NguyenInst. f. Math. u. Phys.Coudraystr. 13BD-99423 Weimar, Deutschlande-mail: [email protected]

[email protected]





Fractional Clifford Analysis

Uwe Kahler and Nelson Vieira

Abstract. In this paper we present the basic tools of a fractional functiontheory in higher dimensions by means of a fractional correspondence to theWeyl relations. A Fischer decomposition, Almansi decomposition, fractionalEuler and Gamma operators, monogenic projection, and basic fractional ho-mogeneous powers will be constructed.

Mathematics Subject Classification (2010). Primary: 30G35. Secondary: 26A33;30A05; 31B05; 30G20.

Keywords. Fractional monogenic polynomials; Fischer decomposition; Almansidecomposition; fractional Dirac operator; Caputo derivatives.

1. Introduction

The use of fractional calculus in mathematical modeling has become popular inrecent years. This popularity arises naturally because on the one hand differentproblems can be considered in the framework of fractional derivatives like, forexample, in optics and quantum mechanics, and on the other hand fractionalcalculus gives us a new degree of freedom which can be used for more completecharacterization of an object or as an additional encoding parameter. Fractionalcalculus, for example, is used for phase retrieval [2], signal characterization [5],space-variant filtering [1], encryption [15], watermaking [10], and creation of neuralnetworks [4].

The connections between fractional calculus and physics are, in some sense,relatively new one but, and more important for the community, a subject of stronginterest. In [17] the author proposed a fractional Dirac equation of order 2/3 andestablished the relation between the corresponding γμ

α-matrix algebra and gener-alized Clifford algebras. This approach was generalized in [18], where the authorfound that relativistic covariant equations generated by taking the nth root of thed’Alembert operator are fractional wave equations with an inherent SU(n) sym-metry. It is clear that the study of fractional problems is a subject of current andstrong investigations, in particular, the study of the fractional Dirac operator due

Switzerland

192 U. Kahler and N. Vieira

to its physical and geometrical interpretations. Physically, this fractional differen-tial operator is related with some aspects of fractional quantum mechanics suchas the derivation of the fractal Schrodinger type wave equation, the resolution ofthe gauge hierarchy problem, and the study of super-symmetries. Geometrically,the fractional classical part of this operator may be identified to the scalar cur-vature in Riemannian geometry. The major problem with most of the fractionalapproaches treated is the presence of non-local fractional differential operators.Furthermore, the adjoint of a fractional differential used to describe the dynamicsis non-negative itself. Other complicated problems arise during the mathematicalmanipulations, as the appearance of a very complicated rule which replaces theLeibniz rule for product of functions in the case of the classic derivative. Also wehave a lack of any sufficiently good analogue of the chain rule. It is important toremark that there several definitions for fractional derivatives (Riemann–Liouville,Caputo, Riesz, Feller,. . . ), however there are not many of these allow our approach.For the purposes of this work, the definition of fractional derivatives in the senseof Caputo is the most appropriate and applicable.

Although these difficulties create problems in the establishment of a fractionalClifford analysis, there is one approach which can be relatively easy adapted. Overthe last decades F. Sommen and his collaborators developed a method for estab-lishing a higher dimension function theory based on the so-called Weyl relations[7, 8, 9]. In more restrictive settings it is nowadays called Howe dual pair technique(see [16]). Its focal point is the construction of an operator algebra (classicallyosp(1|2)) and the resulting Fischer decomposition.

The traditional Fischer decomposition in harmonic analysis yields an orthog-onal decomposition of the space Pl of homogeneous polynomials on Rd of givenhomogeneity l in terms of spaces of harmonic homogeneous polynomials. In clas-sical continuous Clifford analysis one obtains a refinement yielding an orthogonaldecomposition with respect to the so-called Fischer inner product of homogeneouspolynomials, given by

〈P (x), Q(x)〉 = Sc[P (∂x) Q(x)

],

in terms of spaces of monogenic polynomials, i.e., null solutions of the consideredDirac operator (see [9]). Here, the notation Sc[·] stands for taking the scalar part ofa Clifford algebra-valued expression, while P (∂x) is a differential operator obtainedby replacing in the polynomial P each variable xj by the corresponding partialderivative ∂xj and applying Clifford conjugation. This Fischer inner product resultsfrom a duality argument, called Fischer duality, between the algebra of vectorvariables and the algebra of operators. Generalizations as well as refinements ofthe Fischer decomposition in other Clifford analysis frameworks can be found, forexample, in [6, 7, 8, 11, 14, 16].

The aim of this paper is to present a Fischer decomposition, when consideringthe fractional Dirac operator defined via fractional Caputo derivatives, where thefractional parameter α belongs to the interval ]0, 1[. We remark that the cases

Fractional Clifford Analysis 193

where α is outside this range can be reduced to the considered one. In fact, forα ∈ R we have that α = [α] + α, with [α] the integer part of α and α ∈]0, 1[. Tothis end we have to establish the fractional Weyl decomposition and the notionof fractional homogeneous polynomials. As an example we can consider α = 1

2 inour approach which allows us to establish a proper factorization of the transportoperator ∂t−D, whereD represents the Dirac operator and ∂t the partial derivativewith respect to the variable t (see [3]).

The outline of the paper reads as follows. In the Preliminaries we recall somebasic facts about Clifford analysis, fractional Caputo derivatives and fractionalDirac operators. In Section 3 we introduce the corresponding Weyl relations forthis fractional setting and we introduce the notion of a fractional homogeneouspolynomial. Moreover, we present the main result of this work, namely, the frac-tional correspondence to the Fischer decomposition and its extension to a fractionalAlmansi decomposition. In the end of the paper we construct the projection of agiven fractional homogeneous polynomial into the space of fractional homogeneousmonogenic polynomials. We also calculate the dimension of the space of fractionalhomogeneous monogenic polynomials.

2. Preliminaries

We consider the d-dimensional vector space Rd endowed with an orthonormalbasis {e1, . . . , ed}. We define the universal real Clifford algebra R0,d as the 2d-dimensional associative algebra which obeys the multiplication rules eiej + ejei =−2δi,j. A vector space basis for R0,d is generated by the elements e0 = 1 and eA =eh1 · · · ehk

, where A = {h1, . . . , hk} ⊂ M = {1, . . . , d}, for 1 ≤ h1 < · · · < hk ≤ d.The Clifford conjugation is defined by 1 = 1, ej = −ej for all j = 1, . . . , d, and

we have ab = ba. An important property of algebra R0,d is that each non-zero

vector x ∈ Rd1 has a multiplicative inverse given by x

||x||2 . Now, we introduce the

complexified Clifford algebra Cd as the tensor product

C⊗ R0,d ={w =

∑AwAeA, wA ∈ C, A ⊂M

},

where the imaginary unit i of C commutes with the basis elements, i.e., iej = ejifor all j = 1, . . . , d. To avoid ambiguities with the Clifford conjugation, we denotethe complex conjugation mapping a complex scalar wA = aA + ibA, with realcomponents aA and bA, onto wA = aA − ibA by �. The complex conjugation

leaves the elements ej invariant, i.e., e�j = ej for all j = 1, . . . , d. We also have a

pseudonorm on C viz |w| :=∑A |wA| where w =∑

A wAeA, as usual. Notice alsothat for a, b ∈ Cd we only have |ab| ≤ 2d|a||b|. The other norm criteria are fulfilled.

A Cd-valued function f over Ω ⊂ Rd1 has representation f =

∑A eAfA,

with components fA : Ω → C. Properties such as continuity will be understood

component-wisely. Next, we recall the Euclidean Dirac operator D =∑d

j=1 ej ∂xj ,

which factorizes the d-dimensional Euclidean Laplacian, i.e.,

D2 = −Δ = −∑d

j=1∂2xj.

A Cd-valued function f is called left-monogenic if it satisfies Du = 0 on Ω (resp.right-monogenic if it satisfies uD = 0 on Ω). For more details about Cliffordalgebras and monogenic function we refer to [9].

An important subspace of the real Clifford algebra R0,d is the so-called spaceof paravectors Rd

1 = R⊕

Rd, being the sum of scalars and vectors. An an element

x = (x0, x1, . . . , xd) of Rd will be identified by x = x0 + x, with x =∑d

i=1 eixi.From now until the end of the paper, we will consider paravectors of the formxα = xα

0 + xα, where

xαj =

⎧⎪⎨⎪⎩exp(α ln |xj |); xj > 0

0; xj = 0

exp(α ln |xj |+ iαπ); xj < 0

,

with 0 < α < 1, and j = 0, 1, . . . , d.The fractional Dirac operator will correspond to the fractional differential

operator Dα =∑d

j=1 ejC+∂

αj , where

C+∂

αj is the fractional Caputo derivative with

respect to xαj defined as (see [13])

(C+∂

αj f)(xα) =

1

Γ(1− α)

∫ xαj

0

1

(xαj − u)α

f ′u(x

α1 , . . . , x

αj−1, u, x

αj+1, . . . , x

αn) du.

(2.1)A Cn-valued function f is called fractional left-monogenic if it satisfiesDαu = 0 onΩ (resp. fractional right-monogenic if it satisfies uDα = 0 on Ω). For more detailsabout fractional calculus and applications we refer [13]. We remark that due to thedefinition of the fractional Caputo derivative (2.1) we have that Dαc = 0, where cdenotes a constant, i.e., a fractional monogenic function.

3. Weyl relations and fractional Fischer decomposition

The aim of this section is to provide the basic tools for a function theory for thefractional Dirac operator.

3.1. Fractional Weyl relations

Here we introduce the fractional correspondence, via Caputo derivatives, of theclassical Euler and Gamma operators. Moreover, we will show that the two naturaloperatorsDα and xα, considered as odd elements, generate a finite-dimensional Liesuper-algebra in the algebra of endomorphisms generated by the partial Caputoderivatives, the basic vector variables xα

j (seen as multiplication operators), andthe basis of the Clifford algebra ej . Furthermore, we will introduce the definitionof fractional homogeneous polynomials.

In order to achieve our aims, we will use some standard technique in higherdimensions, namely we study the commutator and the anti-commutator betweenxα and Dα. We start proposing the following Weyl relations[

C+∂

αi , x

αi

]= C

+∂αi xα

i − xαi

C+∂

αi =

απ

sin(απ) Γ(1− α)=: Kα, (3.1)

with i = 1, . . . , d, 0 < α < 1, and C+∂

αi is the fractional Caputo derivative (2.1).

This leads to the following relations for our fractional Dirac operator

{Dα, xα} = Dαxα + xαDα = −2Eα −Kαd, (3.2)

[xα, Dα] = xαDα −Dαxα = −2Γα +Kαd, (3.3)

where Eα, Γα are, respectively, the fractional Euler and Gamma operators. Theexpressions for Eα and Γα are

Eα =

d∑i=1

xαi

C+∂

αi ,

Γα =∑i<j

eiej(xαi

C+∂

αj − C

+∂αj xα

i ).

(3.4)

This can be easily checked by

{Dα, xα} =d∑

i=1

d∑j=1

(eiej

C+∂

αi xα

j + ejei xαj

C+∂

αi

)= −

d∑i=1

(C+∂

αi xα

i + xαi

C+∂

αi

)= −

d∑i=1

(2xα

iC+∂

αi +Kα

)= −2Eα −Kαd,

and

[xα, Dα] =

d∑i=1

d∑j=1

(eiej xα

iC+∂

αj − ejei

C+∂

αj xα

i

)=

d∑i�=j

(eiej xα

iC+∂

αj − ejei

C+∂

αj xα

i

)− d∑i=1

(xαi

C+∂

αi − C

+∂αi xα

i

)= −2

d∑i<j

eiej(xαi

C+∂

αj − C

+∂αj xα

i )−d∑

i=1

(−Kα)

= −2Γα +Kαd.


From relations (3.4) we derive, via straightforward calculations, the following re-lations

Eα + Γα = −xαDα, [Eα,Γα] = 0,

[xα,Eα] = −Kαxα, [Dα,Eα] = KαD

α,

{xα, xα} = −2|xα|2, {Dα, Dα} = 2Δ2α,

[xα, |xα|2] = 0, [xα,Δ2α] = −2KαDα,

[Dα, |xα|2] = 2Kαxα, [Dα,Δ2α] = 0,

[Eα,Δ2α] = −2KαΔ2α, [Eα, |xα|2] = 2Kα|xα|2,

[|xα|2,Δ2α] = −4Kα

(Eα +

Kαd

2

),

(3.5)

where Δ2α = −DαDα. The previous relations show that we have a finite-dimen-sional Lie superalgebra generated by xα and Dα, isomorphic to osp(1|2). Indeed,the normalization

Hα =1

2

(Eα +

Kαd

2

),

(Eα)+ =1

2Kα|xα|2, (Eα)− = −1

2KαΔ

2α,

(Fα)+ =1

2√2iKαx

α, (Fα)− =1

2√2iKαD

α,

leads to the standard commutation relations for osp(1|2) (see [12])

[Hα, (Eα)±] = ±Kα(Eα)±, [(Eα)+, (Eα)−] = 2K3

αHα,

[Hα, (Fα)±] = ±1

2Kα(F

α)±, {(Fα)+, (Fα)−} = 1

2K2

αHα,

[(Eα)±, (Fα)∓] = −K2α(F

α)±, {(Fα)±, (Fα)±} = ±1

2Kα(E

α)±.

(3.6)

Now we introduce the definition of fractional homogeneity of a polynomial bymeans of the fractional Euler operator.

Definition 3.1. A polynomial P is called fractional homogeneous of degree l ∈ N0,

if and only if EαP = Kl,α l P , where Kl,α = α Γ(αl)Γ(1+α(l−1)) .

We remark that from the previous definition the basic fractional homoge-neous powers are given by (xα)β = (xα

1 )β1 · · · (xα

d )βd , with l = β1 + · · · + βd. In

combination with the third relation in (3.5)

[xα,Eα] = −Kαxα,

this definition also implies that the multiplication of a fractional homogeneouspolynomial of degree l by xα, will result in a fractional homogeneous polynomialof degree l+1, and thus may be seen as a raising operator. Moreover, we can also


ensure that for a fractional homogeneous polynomial Pl of degree l, DαPl is a frac-

tional homogeneous polynomial of degree l− 1. Furthermore, Weyl relations (3.1)will now enable us to construct fractional homogeneous polynomials, recursively.

3.2. Fractional Fischer decomposition

In the present fractional context, a Fischer inner product of two fractional homo-geneous polynomials P and Q would have the following form

〈P (xα), Q(xα)〉 = Sc[P (∂xα) Q(xα)

], (3.7)

where ∂xα represents C+∂

αi , i.e., the fractional Caputo derivative (2.1) with respect

to xj , and P (∂xα) is a differential operator obtained by replacing in the polynomialP each variable xj by the corresponding fractional Caputo derivative. From (3.7)we have that for any polynomial Pl−1 of homogeneity l − 1 and any polynomialQl of homogeneity l

〈xα Pl−1, Ql〉 = 〈Pl−1, DαQl〉 . (3.8)

This fact allows us to prove the following result:

Theorem 3.2. For each l ∈ N0 we have Πl = Ml + xα Πl−1, where Πl denotesthe space of fractional homogeneous polynomials of degree l and Ml denotes thespace os fractional monogenic homogeneous polynomials of degree l. Furthermore,the subspaces Mk and xαΠl−1 are orthogonal with respect to the Fischer innerproduct (3.7).

Proof. Since

Πl = xα Πl−1 + (xα Πl−1)⊥,

it suffices to prove that (xαΠl−1)⊥ = Ml−1. To this end, assume that, for some

Pl ∈ Πl we have

〈xα Pl−1, Pl〉 = 0, for all Pl−1 ∈ Πl−1.

From (3.8) we then have that

〈Pl−1, DαPl〉 = 0, for all Pl−1 ∈ Πl−1.

As DαPl ∈ Πl−1 we obtain that DαPl = 0, or that Pl ∈ Ml. This means that(xα Πl−1)

⊥ ⊂ Ml−1. Conversely, take Pl ∈ Ml. Then we have, for any Pl−1 ∈Πl−1, that

〈xα Pl−1, Pl〉 = 〈Pl−1, DαPl〉 = 〈Pl−1, 0〉 = 0,

from which it follows thatMl−1 ⊂ (xα Πl−1)⊥, and thereforeMl−1 = (xα Πl−1)

⊥.�

As a result we obtain the fractional Fischer decomposition with respect tothe fractional Dirac operator Dα.


Theorem 3.3. Let Pl be a fractional homogeneous polynomial of degree l. Then

Pl = Ml + xα Ml−1 + (xα)2 Ml−2 + · · ·+ (xα)l M0, (3.9)

where each Mj denotes the fractional homogeneous monogenic polynomial of de-gree j.

The polynomials represented in (3.9) are orthogonal to each other with re-spect to the Fischer inner product (3.7). This a consequence of the construction ofthe fractional Euler operator Eα, and in particular of (3.2) and (3.3). Moreover,from the previous theorem we have the following direct extension to the fractionalcase of the Almansi decomposition:

Theorem 3.4. For any fractional polyharmonic polynomial Pl of degree l ∈ N0 ina starlike domain D in Rd with respect to 0, i.e., Δ2αPl = 0, there exist uniquelyfractional harmonic functions P0, P1, . . . , Pl−1 such that

Pl = P0 + |xα|2P1 + · · ·+ |xα|2(l−1)Pl−1, ∀xα ∈ D.

3.3. Explicit formulae

In this subsection we obtain an explicit formula for the projection πM(Pl) of a givenfractional homogeneous polynomial Pl into the space of fractional homogeneousmonogenic polynomials. We start proving the following auxiliary result:

Theorem 3.5. For any fractional homogeneous polynomial Pl and any positive in-teger s, we have:

Dα(xα)sPl = gs,l(xα)s−1Pl + (−1)s(xα)sDαPl,

where g2k,l = −2kKα and g2k+1,l = −(2(kKα +Kl,αl) +Kαd).

Proof. The proof follows, by induction and making straightforward calculations,from the commutation between Dα and (xα)s using the relations

Dαxα = −2Eα −Kαd− xαDα, Eαxα = xαEα +Kαxα. �

Let us now compute an explicit form of the projection πM(Pl).

Theorem 3.6. Consider the constants cj,l defined by

c0,l = 1, cj,l =(−1)j

(2 [j/2])!!∏[j/2]

i=0g2i+1,l−(2i+1)

,

where j = 1, . . . , l and [·] represents the integer part. Then the map πM given by

πM(Pl) := Pl + c1,lxαDαPl + c2,l(x

α)2(Dα)2Pl + · · ·+ cl,l(xα)l(Dα)lPl

is the projection of the fractional homogeneous polynomial Pl into the space offractional homogeneous monogenic polynomials.


Proof. Let us consider the linear combination

r = a0Pl + a1xαDαPl + a2(x

α)2(Dα)2Pl + · · ·+ ak(xα)l(Dα)lPl,

with a0 = 1. If there are constants aj , j = 1, . . . , l, such that r ∈ Ml, then r isequal to πM(Pl). Indeed, we know that Pl =Ml ⊕ xαPl−1 and

r = Pl +Ql−1, with Ql−1 =l∑

i=1

ai(xα)i(Dα)iPl.

Applying Theorem 3.5, we get

0 = Dα(πM(Pl))

= DαPl + a1DαxαDαPl + a2D

α(xα)2(Dα)2Pl

+ · · ·+ alDα(xα)l(Dα)lPl

= (1 + a1g1,l−1)DαPl + ((−1)1a1 + a2g2,l−2)x

α(Dα)2Pl

+ ((−1)2a2 + a3g3,l−3)(xα)2(Dα)3Pl

+ · · ·+ ((−1)l−1al−1 + algl,0)(xα)l−1(Dα)lPl.

Hence if the relation (−1)j−1aj−1 + aigj,l−j = 0 holds for each j = 1, . . . , l, thenthe function r is fractional monogenic. By induction we get

aj =(−1)j

(2 [j/2])!!∏[j/2]

i=0g2i+1,l−(2i+1)

.

�Theorem 3.7. Each polynomial Pl can be written in a unique way as

Pl =

l∑j=0

(xα)jMl−j(Pl),

where

Ml−j(Pl) = c′j

j∑n=0

cj,l−n(xα)n(Dα)n(Dα)l−jPl, j = 0, . . . , l,

and the coefficients c′j are defined by

c′j =(−1)j

(2 [j/2])!!∏[j/2]

i=0g2i+1,l−(2i+1)

.

Proof. We know that for any Pl, there is a unique decomposition

Pl = (xα)lM0 + (xα)l−1M1 + · · ·+ xαMl−1 +Ml,

where Ml−j ∈ Ml−j , with j = 0, . . . , l. To compute a component Ml−j explicitly,we apply (Dα)j to both sides of the previous equality

(Dα)jPl = (Dα)j(xα)lM0 + (Dα)j(xα)l−1M1

+ · · ·+ (Dα)j(xα)j+1Ml−j−1 + (Dα)j(xα)jMl−j .


The summands on the right-hand side belong, in turn, to the spaces

(xα)l−jM0, (xα)l−j−1M1, · · · xαMl−j−1, Ml−j .

Hence, (Dα)j(xα)jMl−j is equal to πM((Dα)jPl). We can now use the expressionfor the harmonic projection proved above. So to get our result, it is sufficient toshow that

(Dα)j(xα)jMl−j =1

cjMj .

By Theorem 3.5, we get by induction that c′j = aj , and, therefore, the proof isfinished. �

As a consequence, we also obtain the dimension of the space of fractionalhomogeneous monogenic polynomials of degree l. Indeed, from the Fischer decom-position (3.9) we get

dim(Ml) = dim(Πl)− dim(Πl−1),

with the dimension of the space of fractional homogeneous polynomials of degreel given by

dim(Πl) =(k + d− 1)!

k! (d− 1)!.

This leads to the following theorem:

Theorem 3.8. The space of fractional homogeneous monogenic polynomials of de-gree l has dimension

dim(Mk) =(l + d− 1)!− l(l + d− 2)!

l! (d− 1)!=

(l + d− 2)!

l! (d− 2)!.

Acknowledgment

This work was supported by Portuguese funds through the CIDMA – Center forResearch and Development in Mathematics and Applications, and the PortugueseFoundation for Science and Technology (“FCT – Fundacao para a Ciencia e aTecnologia”), within project PEst-OE/MAT/UI4106/2014.

References

[1] O. Akay and G.F. Boudreaux-Bartels, Fractional convolution and correlation viaoperator methods and an application to detection of linear FM signals, IEEE Trans.Signal Processing, 49, No. 5, (2001), 979–993.

[2] T. Alieva, M.J. Bastiaans and L. Stankovic, Signal reconstruction from two closefractional Fourier power spectra, IEEE Trans. Signal Processing, 51, No. 1, (2003),112–123.

[3] V. Agoshkov, Boundary value problems for transport equations, Modeling and Sim-ulation in Science, Engineering and Technology, Birkhauser, Boston, MA, 1998.

[4] B. Barshan and B. Ayrulu, Comparative analysis of different approaches to targetdifferentiation and localization with sonar, Pattern Recognition, 36, No. 5, (2003),1213–1231.


[5] M.J. Bastiaans and T. Alieva, Wigner distribution moments measured as intensitymoments in separable first-order optical systems, EURASIP Journal on Applied Sig-nal Processing, 2005, No. 10, (2005), 1535–1540.

[6] G. Bernardes, P. Cerejeiras and U. Kahler, Fischer decomposition and Cauchy kernelfor Dunkl-Dirac operators, Adv. Appl. Clifford Algebr., 19, No. 2, (2009), 163–171.

[7] H. De Bie and F. Sommen, Fischer decompositions in superspace, in: Function spacesin complex and Clifford analysis, Le Hung Son et al. (eds.), National UniversityPublishers, Hanoi, 2008, 170–188.

[8] H. De Ridder, H. De Schepper, U. Kahler and F. Sommen, Discrete function theorybased on skew Weyl relations, Proc. Am. Math. Soc., 138, No. 9, (2010), 3241–3256.

[9] R. Delanghe, F. Sommen and V. Soucek, Clifford algebras and spinor-valued func-tions. A function theory for the Dirac operator, Mathematics and its Applications,Vol. 53, Kluwer Academic Publishers, Dordrecht etc., 1992.

[10] I. Djurovic, S. Stankovic and I. Pitas, Digital watermarking in the fractional Fouriertransformation domain, Journal of Network and Computer Applications, 24, No. 2,(2001), 167–173.

[11] D. Eelbode, Stirling numbers and spin-Euler polynomials, Exp. Math., 16, No. 1,(2007), 55–66.

[12] L. Frappat, A. Sciarrino and P. Sorba, Dictionary on Lie algebras and superalgebras,Academic Press, San Diego, CA, 2000.

[13] A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and applications of fractionaldifferential equations, North-Holland Mathematics Studies 204, Elsevier, Amster-dam, 2006.

[14] H.R. Malonek and D. Pena-Pena, Fischer decomposition by inframonogenic func-tions, Cubo, 12, No. 2, (2010), 189–197.

[15] N.K. Nishchal, J. Joseph and K. Singh, Fully phase encryption using fractionalFourier transform, Optical Engineering, 42, No. 6, (2003), 1583–1588.

[16] B. Ørsted, P. Somberg and V. Soucek, The Howe duality for the Dunkl version ofthe Dirac operator, Adv. Appl. Clifford Algebr., 19, No. 2, (2009), 403–415.

[17] A. Raspini, Simple solutions of the fractional Dirac equation of order 2/3, PhysicaScripta, 64, No. 1, (2001), 20–22.

[18] P.J. Zavada, Relativistic wave equations with fractional derivatives and pseudodiffer-ential operators, J. Appl. Math., 2, No. 4, (2002), 163–197.

Uwe Kahler and Nelson VieiraCIDMA – Center for Research and Development

in Mathematics and ApplicationsDepartment of Mathematics, University of AveiroCampus Universitario de Santiago3810-193 Aveiro, Portugale-mail: [email protected]

[email protected]





Spectral Properties of Differential Equationsin Clifford Algebras

Yakov Krasnov

Abstract. The aim of this work is to establish a number of elementary proper-ties about the topology and algebra of real quadratic homogeneous mappingsand Riccati type ODEs occurring in Clifford Algebras. We construct an exam-ple of the quadratic vector field to show the impact of the spectral propertieson qualitative theory.

Mathematics Subject Classification (2010). Primary 17A01, 34G20, 11E04,35E20; Secondary 35F05.

Keywords. Bilinearity, differential equations in algebra.

1. Introduction

In mathematics, the term linearization refers to finding a linear approximation of anonlinear model. This essentially allows us to exploit methods of linear operators(linear algebra). Similarly, the term bilinearization refers to finding a quadraticapproximation of a nonlinear model, which in turn allows the use of the methodsof nonassociative commutative binary algebra.

1.1. Linearization in dynamical systems

It is well known that in the study of dynamical systems linearization is a powerfulmethod for assessing the local stability of an equilibrium point of a system ofnonlinear differential equations or discrete dynamical systems. However, sometimesthe qualitative investigations of the system require that we consider the quadraticterms (bilinearization) as well.

This paper is the extended version of a talk presented in the session of the 9th ISAAC Conference

held in Krakow in 2013 titled “Clifford and Quaternionic Analysis” and organized by I. Sabadini,S. Bernstein and F. Sommen.

Switzerland

204 Y. Krasnov

In the study of dynamical systems, the Grobman–Hartman theorem, calledalso the “linearization theorem”, is a result about the local behavior of a dynam-ical system in a neighborhood of a hyperbolic equilibrium point (meaning that noeigenvalue of the linearization has its real part equal to zero).

Basically the theorem states that the behavior of a dynamical system nearthe hyperbolic equilibrium point is qualitatively the same as the behavior of itslinearization.

Therefore, the local dynamics near a hyperbolic equilibrium is completelydetermined by its linearization.

In particular,

• if real parts of all eigenvalues of the Jacobian matrix are negative, then theequilibrium is stable,

• if they are all positive, then the equilibrium is unstable,• if the values are of mixed signs, then the equilibrium is a saddle,• any conjugate pair of complex eigenvalues indicates a spiral.

If, however, the equilibrium is non-hyperbolic, then providing an effectivestability analysis, it constitutes a problem of great complexity. In such a case, itis natural to consider the bilinearization.

1.2. Bilinearization

Denote by x∗ an equilibrium point of a vector field V (x). Then

V (x) = L(x− x∗) +Q(x− x∗) + higher terms with respect to ||x− x∗||2, (1.1)

where L : Rn → Rn is a linearization of V (x) near x∗ and Q : Rn → Rn is a qua-dratic homogeneous map obtained as the second term in an approximation of V (x).

In order to take advantage of appropriate algebraic methods, define the sym-metric bilinear map B(x, y) : Rn × Rn → Rn, associated with Q(x) via the polar-ization identity:

B(x, y) =1

2[Q(x+ y)−Q(x)−Q(y)], ∀x, y ∈ Rn, (1.2)

or, equivalently, by using the Euler formula

B(x, y) =1

2J(x)y, ∀x, y ∈ Rn, (1.3)

where J(x) is the Jacobian matrix of the quadratic map Q(x) at the point(x− x)∗ ∈ Rn.

Note that the use of formulas (1.1)–(1.3) is the essence of the “bilinearization”procedure for a vector field V . This formula allows us to canonically assign to Va nonassociative algebra A = (Rn, ◦) with multiplication

x ◦ y = B(x, y). (1.4)

Several questions relevant to a qualitative study of the field V give rise to a naturalquadratic form with which one can associate the corresponding Clifford algebraCl(V,Q).

Spectral Properties of DEs 205

1.3. Spectral invariants

Given a polynomial vector field V : Rn → Rn, denote by Nul(V ) the set of allstationary points (possibly complex) of the vector field V . Take x∗ ∈ Nul(V ).Then (1.1) takes place with linearization Lx∗ and bilinearization Bx∗ .

Definition 1.1. The collection of eigenvalues of Lx∗ will be called the primaryspectrum of the vector field V at x∗ ∈ Nul(V ).

Denote by Fix(Qx∗) the set of all non-zero fixed points of Qx∗ .

Definition 1.2. Let Ax∗ be an algebra associated with the quadratic map Qx∗ viathe multiplication (1.4). In this algebra are defined the Peirce numbers (see [1])as the collection of eigenvalues of the Jacobian matrix of Qx∗ at any point fromFix(Qx∗). In what follows, these numbers will be called the secondary spectrum ofthe vector field V .

Remark 1.3. Obviously, primary and secondary spectra are invariant with respectto the linear change of variables. If necessary, one can construct spectra of higherorder homogeneous part.

It turns out that the Peirce numbers respect some syzygies related to theEuler–Jacobi formula.

Example. Consider the following quadratic vector field V : R2 → R2:

V (x, y) = {x− y + 4x2 − 2y2, x+ y − 2xy}. (1.5)

Then there exist four equilibria [0, 0], [0,− 12 ], [

12 ,

12 ], [− 1

2 ,12 ]. The non-zero equi-

libria are hyperbolic. Therefore, they can be completely studied using the primaryspectrum only.

Consider the origin, which, obviously, is not hyperbolic.The quadratic term of the expansion of V at the origin is:

Q{0,0}(x, y) = {4x2 − 2y2, −2xy}. (1.6)

Clearly, Q{0,0} has also three fixed points: x1 = [ 14 , 0], x2,3 = [− 12 ,±

√32 ]. After

straightforward calculation we obtain the following Peirce numbers:

μ1 = −1/2; μ2,3 = −5. (1.7)

Also the Peirce numbers respect the following syzygy:

1

μ1 − 1+

1

μ2 − 1+

1

μ3 − 1+ 1 = 0. (1.8)

1.4. Subject of the paper

In this note, we are mainly concerned with studying several connections of primaryand secondary spectra to the qualitative properties of the vector field in question.In our opinion, the following problems are interesting:

• To study a connection between topological/geometric properties of ODEsand PDEs with the properties of an underlying algebra.

206 Y. Krasnov

• To explain the special role of Clifford algebras in differential equations basedon unitality and associativity.

• To extend the known results in Clifford analysis to algebras that are isotopyequivalent to Clifford algebras.

• To clarify the role of syzygies between Peirce numbers as found in dynamicalsystems.

2. Algebras and DEs

Remark 2.1. (i) If V is the polynomial vector field, then by Bezout’s theorem thereare no more than 2n − 1 non-zero fixed points of the homogeneous quadratic mapin Rn or infinitely many.

(ii) Clearly, if z ∈ Cn is an equilibrium for Q, then so is λz for any λ ∈ C.Denote by C[z] the (complex) line of equilibria of Q generated by some (non-zero)z ∈ Cn.

(iii) As is well known (see, for example, [10]), the set Fix(Q) consists of finitelymany connected components of (complex) fixed points. Also, all equilibria Nul(Q)constitute one connected component that is a union of a collection of complexplanes passing through the origin.

Also a spectral invariant is a quantity which is determined by the spectrum ofthe quadratic map, and two quadratic maps are called isospectral, if their spectra,counting multiplicities, coincide.

Given the set P ∈ Cn. The following four problems were studied in [7]:

Problem A (existence). What are the algebraic conditions on the set P providingthe existence of a polynomial map V with bilinearization Q at stationary pointx∗ ∈ Rn such that P = Fix(Q)?

Problem B (uniqueness). Assume a vector field V (x) required in Problem A doesexist. Under which conditions on P is such a V unique (up to a linear change ofvariables)?

Problem C (extension). Let P,N be such that Problem A is solvable for somequadratic map f . How can one extend the sets P,N to the spectrum of f withoutgiving an explicit construction of f (observe that an explicit construction of f canbe a problem of formidable complexity)?

Problem D (topological and dynamical properties). How are the topological anddynamical properties of a vector field V determined by its spectrum?

2.1. Linear constant coefficient ODEs as ODEs in a unary algebra

Given a unary algebra A = (Rn, ) with unary multiplication rule,

ei =

n∑j=1

ajiej , i = 1, . . . , n. (2.1)


Any linear constant coefficient ODE x = Lx can be rewritten as

x(t) = x(t), (2.2)

where x(t) = x1(t)e1 + · · ·+ xn(t)en and Lx = x in A.

Formula (2.2) indicates the parallelism between unary algebras and linearODEs. The universal language allowing us to study both problems is the spectraltheory for linear operators.

2.2. First-order PDEs and ODEs in binary algebras

Given a binary algebra A = (Rn, ◦) with the multiplication rule:

ei ◦ ej =n∑

j=1

akijek, i, j = 1, . . . , n. (2.3)

Then, using this rule, any quadratic ODE x = Q(x) with homogeneous quadraticmap Q : Rn → Rn can be rewritten as a Riccati equation in the algebra:

x(t) = x(t) ◦ x(t). (2.4)

Also, any first-order linear PDE with constant coefficients can be thought ofas the Dirac equation in a binary algebra as follows:

D ◦ u(x) = 0, (2.5)

where D = ∂x1e1 + · · ·+ ∂xnen, u(x) = u1(x)e1 + · · ·+ un(x)en.

Qualitative properties of the Riccati equation (2.4) can be studied in terms ofinvariants of algebra isomorphisms. The universal language for this investigationis primary and secondary spectral theory.

In contrast to the case of the Riccati equation in binary algebra, the structureof solutions to the Dirac equation (2.5) refers to isotopy invariants of A rather thanto the isomorphism [6].

2.3. Complex structure

Usually, by a complex structure one understands a linear map J : Rn → Rn

satisfying J2 = −I, where I is the identity operator (cf. [Gr]). Below we extendthis concept to real commutative algebras.

Definition 2.2. Let A be a real finite-dimensional (in general, nonassociative) al-gebra. By a complex structure in A we mean the existence of a non-trivial solutionto the equation x2 ◦ y = −y, as a generalization of the real equation x2 = −1 inthe unital algebra.

Our study of complex structures is based on the following three observations:

– complex structures are two-dimensional in nature (this motivates our specialinterest in two-dimensional subalgebras);

– in each real two-dimensional algebra, complex structures perform in at mosttwo different ways (cf. Section 2 and Theorems A–C in[2]));

208 Y. Krasnov

– the algebra C of complex numbers is the only real two-dimensional algebrawhere equation x2 ◦ y = −y is soluble for all y.

The presence of complex structures turns out to be responsible for:

– the solubility of polynomial equations with real coefficients (at least of degreethree (see Theorem A in [2]);

– the existence of continuous deformations to C in the class of algebras respect-ing these structures (see Theorem A in [2]);

– the existence of bounded solutions to the Riccati equation in algebras (see [3]);– the ellipticity of the operator of the Dirac operator in algebras (see [6]);

and many other matters that, at first glance, do not have any connection withcomplex structures.

A. Albert [1] considered complex structures for unital division algebras. H.Petersson [9] related Albert isotopy classes to several canonical representatives(the so-called unital hearts). An important class of the unital hearts provide theClifford Algebras.

2.4. Euler–Jacobi formula

Let F (x) = {F1(x) · · ·Fn(x)} be a polynomial vector field and degFi = mi. De-note by SolF the set of roots of a system of n polynomial equations of degreesm1, . . . ,mn in n complex unknowns, F1 = · · · = Fn = 0. Assume that the setSolF contains exactly μ = m1, . . . , mn elements. In this case, the Jacobian of thesystem J = det ∂F

∂x does not vanish on the set SolF . Then, for every polynomial Pof degree less than

∑imi, we have the following Euler–Jacobi formula:∑

a∈SolF

P (a)

J(a)= 0. (2.6)

2.5. Invariants and syzygies

The term “syzygies” was introduced in 1853 by Sylvester to denote some rela-tions among invariants. He discovered that various invariants of the same formsometimes satisfy rational relations among them.

Using Bezout’s theorem and [10], we can write the Euler–Jacobi formula forbilinearization Q(x) in (1.1) as follows:

Theorem 2.3 ([7]). Let pi =(x(i)1 , . . . , x

(i)n

)for i = 1, 2, . . . ,M = 2n − 1 be

(complex) fixed points of the quadratic homogeneous polynomial vector field Q(x)in (1.1). Denote by γi the numbers γi = 1/D

(V (x)− Id

)∣∣x=pi

, where D(·) standsfor the Jacobian determinant. Then the following syzygies among pi, γi hold:∑M

i=1γi + (−1)n = 0, (2.7)∑M

i=1γip

αj

i = 0, |αj | = j, j = 1, . . . , n− 1, (2.8)

where α is a multi-index, xα = xα11 · · · · · xαn

n .


Proof. By Bezout’s theorem in the projective space CPn, there are no more than2n solutions to equation Q(x) = λx, (counted according to their multiplicities)provided that the number of solutions is finite. Therefore, either there exist exactly2n − 1 non-proportional non-zero fixed points Q(p) = p or infinitely many.

Consider separately p = 0 and p = 0 for Q(p) = p:

J(0) = D(−Id) = (−1)n, lima→0

P (p)

J(p)= (−1)nP (0). (2.9)

Using these formulas, Theorem 2.3 follows from (2.6). �

3. Isotopy invariants

The notion of isotopy is a natural generalization of an algebra isomorphism and isdefined as follows:

Definition 3.1. Two algebras (G, ◦) and (H, ∗) are called isotopic (or G is an isotopeof H) if there are linear maps K, L, M from G to H , such that K(x) ◦ L(y) =M(x ∗ y) for all x, y.

Clearly, isotopy is an equivalence relation. If K = L = M one obtains analgebraic isomorphism. To determine the properties invariant under isotopy A isa problem of great importance.

4. Clifford algebras and bounded solutions

The following properties of Clifford algebras make them important for the quali-tative study of DEs:

– Unitality – the DEs in unital algebras are of evolution type.– Associativity – it makes clear the structure of the power series solution of theDirac equations.

– The presence of a complex structure is responsible for the existence of abounded solution to the Riccati type ODEs.

Many real life models reduce to DEs in Clifford algebras using isotopy. As is wellknown, any two Clifford algebras of the same dimension can be distinguished bythe signature of the related quadratic form and this leads to allocation of a differentisotopy class of Clifford algebras.

The following theorem relates the unboundedness of a quadratic polynomialdifferential system

x = L(x) +Q(x) (4.1)

to existence of a real fixed point p of the corresponding homogeneous polynomialsdifferential system

y = Q(y) (4.2)

210 Y. Krasnov

where x and y are real n-vectors, L(x) and Q(x) are real n-vector polynomialfunctions, Q(x) is homogeneous of degree 2, and L(x) is linear (cf. [4], p. 71).

Theorem 4.1. If system (4.2) has a real fixed point Hm(p) = p, then system (4.1)is unbounded.

Consider now the Volterra equation in Rn:

xi = Hi(x), Hi(x) = xi

(xi +

∑j �=i

aijxj

), i = 1, 2, . . . , n. (4.3)

All the basis vectors e1, e2, . . . , en are fixed points of the quadratic system(4.3). Any coordinate plane R[xi, xj ] is an invariant set of (4.3). Moreover, ifaij , aji = 1 and aijaji = 1, then the additional fixed point

pij =1− aij

1− aijajiei +

1− aji1− aijaji

ej (4.4)

lies in R[xi, xj ].

Denote by γi, γj , γij the same numbers, as quantities in Theorem 2.3, re-stricted on R[xi, xj ] and computed at the fixed points ei, ej , pij in R[xi, xj ] cor-respondingly. Then by straightforward computations:

γi =1

aji − 1, γj =

1

aij − 1, γij =

1− aijaji(1− aij)(1− aji)

. (4.5)

Using results in [2] we can prove the following

Theorem 4.2. If there exists such an i, j that two of the following three numbersγi, γj or γij are positive then there exists a bounded solution to (4.3).

In contrast to Theorem 4.2 for existence of a bounded solution to (4.3) it isnecessary for an underlying algebra to have a bounded solution:

Theorem 4.3. [2] Let A be a rank three algebra without 2-nilpotents. Then theRiccati equation (4.3) admits a non-trivial solution bounded on the whole axis iffA contains a complete complex structure.

As a consequent result one can obtain that the Riccati equation (4.3) inClifford algebra Clp,q(R) always has a bounded solution.

5. Conclusions

Many well-known differential systems can be considered to be sets of differen-tial equations operating in algebraic terms. As an example, we discuss differentalgebraic operations acting in subspaces of the solution space of a set of DEs.

Actually, the study of complex structures acts as a particular case of the gen-eral approach adopted in this paper. Namely, we are trying to study algebras fromthe viewpoint of the solubility of certain important polynomial equations with realconstant coefficients. Solvability of such equations in algebras leads to the stability


theory for ODE’s (see [2], [3]), linear isotopy (see [1]), to mention a few. At thesame time many algebraic properties (for example, the number of their solutions)are not homotopy invariants and the choice of a suitable language essentially de-pends on the properties themselves. In fact, the choice of an appropriate languagemeans a certain classification of algebras.

Acknowledgment

The author would like to thank Prof. Irene Sabadini for her valuable commentsand suggestions to improve the quality of the paper as well as for her attentionand time.

References

[1] A. Albert, Nonassociative algebras, Ann. of Math. 43, 1942, 685–707.

[2] Z. Balanov and Y. Krasnov, Complex Structures in Real Algebras. I. Two-Dimen-sional Commutative Case, Communications in Algebra, 31(9), (2003), 4571–4609.

[3] Z. Balanov, Y. Krasnov, A. Kononovich, Projective dynamics of homogeneous sys-tems: local invariants, syzygies and global residue theorem, in Proc. of the EdinburghMath. Soc., 55(3), (2012), 577–589.

[4] C.S. Coleman, Systems of differential equations without linear terms, in NonlinearDifferential Equations and Nonlinear Mechanics, Academic Press, (1963), 445–453.

[5] W.A. Coppel, A survey of quadratic systems, J. Differential Equations 2 (1966),293–304.

[6] Y. Krasnov, Properties of ODEs and PDEs in Algebras, Complex Analysis and Op-erator Theory, 7(3), 2013, 623–634.

[7] Y. Krasnov, A. Kononovich and G. Osharovich, On a structure of the fixed point setof homogeneous maps, Discrete & Continuous Dynamical Systems – Series S, AIMSJournal, 6(4), (2013), 1017–1027.

[8] L. Markus, Quadratic differential equations and non-associative algebras, Annals ofMathematics Studies, 45, 1960, 185–213.

[9] H. Petersson, The classification of two-dimensional nonassociative algebras, Resultin Math. 37, 2000, 120–154.

[10] I.R. Shafarevich, Basic Algebraic Geometry I, Springer, (1994) 317.

Yakov KrasnovDepartment of MathematicsBar-Ilan University52900 Ramat-Gan, Israele-mail: [email protected]




Differential Equations in Multicomplex Spaces

Daniele C. Struppa, Adrian Vajiac and Mihaela B. Vajiac

Abstract. We use Ehrenpreis’ Fundamental Principle to give different rep-resentations of holomorphic functions in multicomplex spaces, and we givethe first elements of a theory of constant coefficients differential equations, offinite and infinite order, in the multicomplex setting.

Mathematics Subject Classification (2010). 30A97, 34A20, 35C10.

Keywords. Multicomplex spaces, constant coefficients differential equations.

1. Introduction

Let BCn be the ring of multicomplex numbers (we refer to Section 3 for the spe-cific definitions, but we recall that BC1 is the field of complex numbers and BC2

is the ring of bicomplex numbers). The theory of holomorphic functions from BCn

to itself is by now well developed [6, 15, 27, 28], and it shows in particular thata function F is multicomplex holomorphic if and only if it is a complex holomor-phic map from C2n to itself, and its 2n complex components satisfy a system offirst-order constant coefficients differential equations known as the multicomplexCauchy–Riemann system.

In this paper we extend to the multicomplex setting some results from [24].In particular we will show how to use the Fundamental Principle of Ehrenpreisto give a suitable representation for holomorphic functions in BCn, and in thisway we will recover the idempotent representation for holomorphic multicomplexfunctions.

From this point of view the space of holomorphic multicomplex functions,considered as the kernel of a system of differential equations on the space of holo-morphic maps in suitable dimensions, ends up being an Analytically Uniform Spacein the sense of Ehrenpreis.

As such, we were also able to study ordinary differential equations of infiniteorder in such a space and provide an exponential representation for such solutions.

The plan of the paper is as follows. In Section 2 we recall the fundamentalideas on Analytically Uniform Spaces and we recall the Fundamental Theorem of

Switzerland

214 D.C. Struppa, A. Vajiac and M.B. Vajiac

Ehrenpreis. In Section 3 we introduce bicomplex and more generally multicomplexspaces, and we give the basic properties of holomorphic functions in those settings.The next section is devoted to the use of the Fundamental Principle to obtainexponential representations for such holomorphic functions. The final section showshow to use these same ideas to begin a study of linear constant coefficients ordinarydifferential equations of both finite and infinite order.

2. Analytically uniform spaces

This section is preliminary to the rest of the paper and can be skipped by thereader who is familiar with the work of Ehrenpreis. Following [10], we consider aspace W of (generalized) functions on Rn or Cn. We will say that W is analyticallyuniform (AU) if such a space satisfies the following conditions:

a) The space W is the dual of a locally convex topological space W ′ and thetopology of W is given by uniform convergence on bounded sets of W ′.

b) For all z ∈ C, the exponential function x � exp(ix · z) belongs to W and theresulting map from C toW is complex analytic. Moreover linear combinationsof these exponentials are dense inW . For any S ∈W ′, we can therefore define

its Fourier transform S := S(exp(ix·z)); such function is defined for all z ∈ C,is entire, and it determines S.

c) There exists a family K of continuous positive functions on C, which can

take the value ∞, such that for all F ∈ W ′ and all k ∈ K,|F (z)|k(z)

−→ 0 as

|z| → ∞. Moreover, the sets Nk of functions in W ′ that are bounded by k

form a fundamental system of neighborhoods of the origin in W ′ (i.e., everyNk is a neighborhood of 0 and any neighborhood of 0 contains an Nk).

Any space satisfying conditions a), b), and c) is called an AU-space and K issaid to be an AU-structure on it. The main examples of AU-spaces are the space Oof entire functions, the space E of infinitely differentiable functions, the space D′ ofdistributions, and the spaceD′

F of distributions of finite order. A classical exampleof a space which is surprisingly not AU is the space of real analytic functions. Itis also known that the space of solutions of a system of linear constant coefficientspartial differential equations in any AU-space is still an AU-space.

The reason for the importance of these spaces lies in the fact that solutionsof systems of linear constant coefficients differential equations in these spaces canbe given a very important exponential representation known as the FundamentalPrinciple of Ehrenpreis–Palamodov. Strictly speaking, this is true only for a specialsubset of such spaces (the so-called Product Localizeable Analytically UniformSpaces) but we will not give the additional technical details that are not necessaryfor the rest of this paper. We refer the reader to, e.g., [2, 9, 10, 23].

This Fundamental Principle can be stated, at least in a special case, as follows:

Differential Equations in Multicomplex Spaces 215

Theorem 2.1. Let P1, . . . , Pr be polynomials in n variables, and let D =(− i ∂

∂x1,

. . ., −i ∂∂xn

). Then there exist algebraic varieties V1, . . . , Vt in Cn and differential

operators ∂1, . . . , ∂t with polynomials coefficients, such that every f ∈ C∞(Rn)satisfying the system

P1(D)f = · · · = Pr(D)f = 0, (1)

can be represented as

f(x) =t∑

�=1

∫V�

∂�(eix·z)dν�(z) , (2)

where dν�(z) are suitable Radon measures supported on V�. The collection

V = {(V1, ∂1); . . . ; (Vt, ∂t)}is called a multiplicity variety.

Remark 2.2. If the system (1) above consists of only one equation, then ∂1, . . . , ∂thave constant coefficients. The operators ∂1, . . . , ∂t are called, in Palamodov’s ter-minology [14], Noetherian operators because their construction relies essentiallyon a theorem of M. Noether on a membership criterion for polynomial submod-ules. The nature of the original proof of the Fundamental Principle is essentiallyexistential and therefore the question of the explicit construction of such operatorsis of great interest (see, e.g., [8] and the references therein).

Remark 2.3. As we indicated, the statement above is a special case of the Fun-damental Principle. To begin with, the space of infinitely differentiable functionscan be replaced with any other AU-space, but in that case the representation (2)need to be interpreted in the sense of generalized functions. Most important, forour specific concerns, the system (1) can actually be generalized to be a matrix

system, so that the unknown is not a single function f but actually a vector �f offunctions satisfying a matrix system of differential equations. This is importantto remember because the multicomplex holomorphic functions fall exactly in thiscategory.

3. Multicomplex spaces

Without giving many details (for which we refer the reader to the fairly compre-hensive recent references [6, 12, 27, 28]), we will simply say that the space BCn

of multicomplex numbers is the space generated over the reals by n commutingimaginary units. The algebraic properties of this space and analytic properties ofmulti-complex-valued functions defined on BCn has been studied in [28]. It is worthnoting also that the algebraic properties of the space BC2 and the properties of itsholomorphic functions have been discussed before in [6, 7, 27], using computationalalgebra techniques. Other related references include [4, 16, 17, 18, 19, 20, 21, 22].

In the case of only one imaginary unit, denoted by i1, the space BC1 is theusual complex plane C. Since, in what follows, we will have to work with different


complex planes, generated by different imaginary units, we will denote such aspace also by C(i1), in order to clarify which is the imaginary unit used in thespace itself.

The next case occurs when we have two commuting imaginary units i1 andi2. This yields the bicomplex space BC2. This space has extensively been studiedin [4, 15, 16], and more recently by the authors in [6, 7, 27], where it is referred toas as BC.

Because of the various units in BC2, we have several different conjugationsthat can be defined naturally. Let therefore consider a bicomplex number Z, andlet us write it both in terms of its complex coordinates, Z = z1 + i2z2 withz1, z2 ∈ C(i1), and in terms of its real coordinates Z = x1 + i1y1 + i2x2 + i1i2y2with x�, y� ∈ R. Note that k12 := i1i2 is a hyperbolic unit, i.e., it squares to 1.

For a bicomplex number defined as Z = z1 + i2z2 we mention the existence

of three conjugations in C(i1): we will denote by Zi1, the one corresponding to the

involution i1 �→ −i1, by Zi2, the one corresponding to the involution i2 �→ −i2,

and finally by Zi1i2

, that corresponding to both involutions. All these conjugationshave been carefully described in [6, 7].

A function F : BC2 → BC2 can be written as F = f1 + i2f2, where f1, f2 :BC2 → BC1 = C(i1). Following the notations from [6], we introduce the followingbicomplex differential operators, written in the standard basis of BC2, seen as C2

in the variables z1 and z2:

∂

∂Z:=

1

2

(∂

∂z1− i2

∂

∂z2

),

∂

∂Zi2

:=1

2

(∂

∂z1+ i2

∂

∂z2

),

∂

∂Zi1

:=1

2

(∂

∂z1− i2

∂

∂z2

),

∂

∂Zi1i2

:=1

2

(∂

∂z1+ i2

∂

∂z2

),

(3)

where∂

∂z1,

∂

∂z2are the usual complex derivatives. These differential operators also

have multiple descriptions, for which we refer the reader to [6, 7].

BC2 is not a division algebra, and it has two distinguished zero divisors, e12and e12, which are idempotent, linearly independent over the reals, and mutuallyannihilating with respect to the bicomplex multiplication:

e12 :=1 + i1i2

2, e12 :=

1− i1i22

.

Just like {1, i2}, they form a basis of the complex algebra BC2, which is called theidempotent basis. If we define the following complex variables in C(i1):

β1 := z1 − i1z2, β2 := z1 + i1z2 ,


the idempotent representations for Z = z1 + i2z2 and its conjugates are given by

Z = β1e12 + β2e12, Zi1= β

i12 e12 + β

i11 e12,

Zi2= β2e12 + β1e12, Z

i1i2= β

i11 e12 + β

i12 e12.

In the idempotent representation, the bicomplex differential operators definedabove become:

∂

∂Z=

∂

∂β1e12 +

∂

∂β2e12 ,

∂

∂Zi2

=∂

∂β2e12 +

∂

∂β1e12 ,

∂

∂Zi1

=∂

∂β2i1e12 +

∂

∂β1i1e12 ,

∂

∂Zi1i2

=∂

∂β1i1e12 +

∂

∂β2i1e12 .

The following notion of a bicomplex derivative is introduced:

Definition 3.1. Let Ω be an open set in BC2 and let Z0 ∈ Ω. A function F : Ω →BC2 is called bicomplex derivable at Z0 if the limit

limZ→Z0

(Z − Z0)−1 (F (Z)− F (Z0))

exists, for all Z in Ω such that Z −Z0 is invertible, i.e., it is not a divisor of zero.When the limit exists, we will call it F ′(Z0), and we will say that the function Fhas derivative equal to F ′(Z0) ∈ BC2 at Z0.

Note that the limit in the definition above avoids the divisors of zero in BC2,which are the union of the two ideals generated by e12 and e12, the so-called coneof singularities. The set of zero divisors together with 0 is usually denoted by S0.

Functions which admit bicomplex derivative at each point in their domainare called bicomplex holomorphic, and it can be shown that this is equivalentto require that they admit a power series expansion in Z [15, Definition 15.2].However, there are more equivalent statements of bicomplex holomorphy [27]. Forexample:

Theorem 3.2. Let Ω be an open set in BC2 and let F = f1 + i2f2 : Ω→ BC2 be ofclass C1(Ω). Then F is bicomplex holomorphic on Ω if and only if:

1. f1 and f2 are complex holomorphic in both complex C(i1) variables z1 andz2.

2.∂f1∂z1

=∂f2∂z2

and∂f2∂z1

= −∂f1∂z2

on Ω; these equations are called the complex

Cauchy–Riemann conditions.

Moreover,

F ′ =1

2

∂F

∂Z=

∂f1∂z1

+ i2∂f2∂z1

=∂f2∂z2

− i2∂f1∂z2

,

and F ′(Z) is invertible if and only if the corresponding Jacobian is non-zero.


In the idempotent basis, we have the following characterization of bicomplexholomorphy:

Theorem 3.3. A bicomplex function F : Ω→ BC2 is bicomplex holomorphic if andonly if

F = G1(β1)e12 +G2(β2)e12 ,

where G1 is a complex holomorphic function on the complex domain Ω1 definedby Ω1 · e12 := Ω · e12, and G2 is a complex holomorphic function on the complexdomain Ω2 defined by Ω2 · e12 := Ω · e12.

An immediate consequence of this theorem is that a bicomplex holomorphicfunction F defined on Ω extends to a bicomplex holomorphic function defined on

Ω := Ω1 · e12 +Ω2 · e12, which strictly includes Ω. Domains of this form are calledbicomplex domains.

A further interesting characterization of holomorphicity in BC2 is the follow-ing result from [6]:

Theorem 3.4. Let Ω ⊆ BC2 be an open set and let F : Ω→ BC2 be of class C1 on

Ω. Then F is bicomplex holomorphic if and only if F is Zi1, Z

i2, Z

i1i2-regular, i.e.,

∂F

∂Zi1

=∂F

∂Zi2

=∂F

∂Zi1i2

= 0.

Note that both Theorems 3.2 and 3.4 indicate that holomorphic functionson bicomplex variables can be seen as solutions of overdetermined systems of dif-ferential equations with constant coefficients. We have exploited this particularityin [6, 7, 27].

We now turn to the definition of the multicomplex spaces, BCn, for val-ues of n ≥ 2. These spaces are defined by taking n commuting imaginary unitsi1, i2, . . . , in, i.e., i

2a = −1, and iaib = ibia for all a, b = 1, . . . , n. Since the product

of two commuting imaginary units is a hyperbolic unit, and since the product ofan imaginary unit and a hyperbolic unit is an imaginary unit, we see that theseunits will generate a set An of 2n units, 2n−1 of which are imaginary and 2n−1 ofwhich are hyperbolic units. Then the algebra generated over the real numbers byAn is the multicomplex space BCn which forms a ring under the usual addition andmultiplication operations. As in the case n = 2, the ring BCn can be representedas a real algebra, so that each of its elements can be written as Z =

∑I∈An

ZII,where ZI are real numbers.

In particular, following [15], it is natural to define the n-dimensional multi-complex space as follows:

BCn := {Zn = Zn−1,1 + inZn−1,2

∣∣Zn−1,1, Zn−1,2 ∈ BCn−1}with the natural operations of addition and multiplication. Since BCn−1 can bedefined in a similar way using the in−1 unit and multicomplex elements of BCn−2,


we recursively obtain, at the kth level:

Zn =∑

|I|=n−k

n∏t=k+1

(it)αt−1Zk,I

where Zk,I ∈ BCk, I = (αk+1, . . . , αn), and αj ∈ {1, 2}.Because of the existence of n imaginary units, we can define multiple types

of conjugations. Following [30] we define:

Znil=

∑|I|=n−k

n∑i=k+1

δl,i(−1)αl−1ckZk,I +∑

|I|=n−k

k∑i=1

δl,ickZk,Iil.

Just as in the case of BC2 we have idempotent bases in BCn, that will beorganized at each “nested” level BCk inside BCn as follows. Denote by

ekl :=1 + ikil

2, ekl :=

1− ikil2

.

Consider the following sets:

S1 := {en−1,n, en−1,n},S2 := {en−2,n−1 · S1, en−2,n−1 · S1},

...

Sn−1 := {e12 · Sn−2, e12 · Sn−2}.At each stage k, the set Sk has 2k idempotents. It is possible to immediately verifythe following

Proposition 3.5. In each set Sk, the product of any two idempotents is zero.

We have several idempotent representations of Zn ∈ BCn, as follows.

Theorem 3.6. Any Zn ∈ BCn can be written as:

Zn =

2k∑j=1

Zn−k,jej ,

where Zn−k,j ∈ BCn−k and ej ∈ Sk.

Due to the fact that the product of two idempotents is 0 at each level Sk, wewill have many zero divisors in BCn organized in “singular cones”. The topologyof the space is difficult, but just like in the case of BC2 we can circumvent this byavoiding the zero divisors to define the derivative of a multicomplex function asfollows.

Definition 3.7. Let Ω be an open set of BCn and let Zn,0 ∈ Ω. A function F : Ω→BCn has a multicomplex derivative at Zn,0 if

limZn→Zn,0

(Zn − Zn,0)−1 (F (Zn)− F (Zn,0)) =: F ′(Zn,0) ,

exists whenever Zn − Zn,0 is invertible in BCn.


Just as in the case of BC2, functions which admit a multicomplex derivativeat each point in their domain are called multicomplex holomorphic, and it can beshown that this is equivalent to require that they admit a power series expansionin Zn [15, Section 47].

We will denote by O(BCn) the space of multicomplex holomorphic functions.A multicomplex holomorphic function F ∈ O(Ω), where Ω ⊂ BCn, can be split asF = U + inV, where U, V are holomorphic functions of two BCn−1 variables. Asin the case n = 2, there is an equivalent notion of multicomplex holomorphicity,which is more suitable for our computational algebraic purposes, and the followingtheorem can be proved in a similar fashion as its correspondent for the case n = 2(the differential operators that appear in the statement are defined in detail lateron in the paper, but it is not difficult to imagine their actual definition).

Theorem 3.8. Let Ω be an open set in BCn and let F : Ω → BCn be such thatF = U + inV ∈ C1(Ω). Then F is multicomplex holomorphic if and only if:

1. U and V are multicomplex holomorphic in both multicomplex BCn−1 variablesZn−1,1 and Zn−1,2.

2.∂U

∂Zn−1,1=

∂V

∂Zn−1,2and

∂V

∂Zn−1,1= − ∂U

∂Zn−1,2on Ω; these equations are

called the multicomplex Cauchy–Riemann conditions.

There are several other ways to identify multicomplex holomorphic functionsin BCn, as there are many idempotent representations with respect to each levelSk. For example, for k = n− 1, we obtain the following characterization of BCn-holomorphy:

Theorem 3.9. A multicomplex function F : Ω → BCn is BCn-holomorphic if andonly if

F =∑2n−1

�=1G�(β�)e� ,

where e� ∈ Sn−1 and G� are complex holomorphic functions on the complex do-mains Ω� defined by Ω� · e� := Ω · e�, for all � = 1, . . . , 2n−1.

As before, a multicomplex holomorphic function F defined on Ω extends toa multicomplex holomorphic function defined on

Ω :=∑2n−1

�=1Ω� · e�

which strictly includes Ω. Domains of this form are called multicomplex domains.

4. Representations of bicomplex and multicomplexholomorphic functions

In the theory of bicomplex functions, the complex light cone in two dimensions,i.e., Γ = {(z1, z2)

∣∣ z21 + z22 = 0}, plays a very important role as it coincides withthe set S0 of zero-divisors in BC2 (together with 0). The complex Laplacian plays


a prominent role because it can be factored as the product of two linear operators,one of which is the bicomplex differentiation.

Indeed

ΔC2(i1) = 4∂

∂Z

∂

∂Z† , (4)

where the operators act on BC2-valued functions holomorphic in the sense of thecomplex variables z1, z2. Hence, the theory of BC2-holomorphic functions can beseen as the function theory for C(i1)-complex Laplacian. This factorization allowsus to establish direct relations between BC2-holomorphic functions and complexharmonic functions, that is, null solutions to the operator ΔC2(i).

Denote by Ω→ B2(Ω) the sheaf of bicomplex holomorphic functions (see [6],[7], [27], [28]). Then, if we use the notation SP to denote the sheaf of S solutionsto the differential operator P we have that:

B2(BC) ![O(C2)×O(C2)

]∂Zi2

where we recall that

∂Zi2 :=∂

∂Z i2=

∂

∂z1+ i2

∂

∂z2is the bicomplex differential operator acting on bicomplex functions F =f1+i2f2 by

f1 + i2f2 �→ (∂z1f1 − ∂z2f2, ∂z2f1 + ∂z1f2) .

We now apply the formalism described thoroughly in [5] and applied to the bi-complex case in [6]. This formalism defines an associated matrix representationto a linear partial differential operator and its associated variety in the sense ofEhrenpreis. We obtain the matrix representation of ∂Zi2 as:(

θ1 −θ2θ2 θ1

)and its associated variety as

V = {Θ = (θ1, θ2) ∈ C2∣∣ θ21 + θ22 = 0} .

The variety V splits into a union V1 ∪ V2, where V1 = {θ1 − i1θ2 = 0} andV2 = {θ1 + i1θ2 = 0}. Using the Fundamental Principle for f1(z1, z2), we write:

f1(z1, z2) =

∫θ21+θ2

2=0

ei1(z1,z2)·(θ1,θ2) dθ1 dθ2

=

∫θ1−i1θ2=0

ei1(z1,z2)·(t,i1t) dt+∫θ1+i1θ2=0

ei1(z1,z2)·(t,−i1t) dt

=

∫C(i1)

ei1t(z1−i1z2) dt+

∫C(i1)

ei1t(z1+i1z2) dt

= g1(z1 − i1z2) + g2(z1 + i1z2).

Similarly,

f2(z1, z2) = h1(z1 − i1z2) + h2(z1 + i1z2).


If we write β1 = z1 − i1z2 and β2 = z1 + i1z2, then the first equation in theCauchy–Riemann system is enough to imply

h1(β1) = i1g1(β1),

h2(β2) = −i1g2(β2),

up to a complex constant. Using the two representations for f1, f2, we obtain

F = f1 + i2f2 = g1(β1)(1 + i1i2) + g2(β2)(1 − i1i2)

= 2g1(β1)e12 + 2g2(β2)e12.

The Fundamental Principle therefore captures the fact that in the idempotentrepresentation, bicomplex holomorphic functions have the form above, where g1 =g1(β1) is a holomorphic function of one complex variable β1, and g2 = g2(β2) is aholomorphic function of one complex variable β2 and we recover the familiar resultfound in [6, 7, 15]:

Theorem 4.1. If F is a bicomplex holomorphic function then it can be written as:F = 2g1(β1)e12 + 2g2(β2)e12. where g1, g2 are holomorphic functions of a singlecomplex variable.

The same result is obtained if we use directly the idempotent representation ofbicomplex numbers and functions: write Z = β1e12+β2e12 and F = u1e12+u2e12,where

β1 = z1 − i1z2 and β2 = z1 + i1z2

are C(i1)-complex numbers, and

u1 := f1 − i1f2 and u2 := f1 + i1f2

are complex functions in β1 and β2. Then the operator ∂Z

i2 is given by

∂Z

i2 = ∂β2e12 + ∂β1e12

acting on F = (u1, u2).

The varieties are

V1 = {β1 = 0} and V2 = {β2 = 0},which gives directly the same result as in the standard basis:

F = u1(β1)e12 + u2(β2)e12,

where u1 = 2g1 and u2 = 2g2.

Note that in the idempotent coordinates, we have:

B2(BC) ![O(C2)×O(C2)

]∂β2e12+∂β1

e12.

In an analogous way we can analyze the multicomplex case.


5. Differential equations of finite and infinite order in BC2 and BCn

In this section we show that, quite simply, linear constant coefficients differentialequations on multicomplex holomorphic functions behave exactly as one wouldexpect. We therefore begin with a simple result that gives a representation for thesolutions of such homogeneous equations.

Theorem 5.1. Let P (Z) be a polynomial in BC2, where P (Z) =∑n

k=0 akZk, with

all zeroes of multiplicity 1. The there exist points ν1, . . . , ν�, . . . , νt such that anysolution of the equation P (D)F = 0 where F is a bicomplex holomorphic functionand D = ∂Z can be written as:

F (Z) =

t∑�=1

c�(eν�Z) ,

where ν� are the zeroes of P , and the c� are suitable bicomplex numbers.

If P has zeroes ν1, . . . , ν�, . . . , νt of multiplicities m� respectively, then F be-comes:

F (Z) =t∑

�=1

p�(Z)(eν�Z) ,

where p�(Z) are suitable bicomplex polynomials with degree m� − 1.

Proof. We start by re-writing the polynomial P (Z) and bicomplex holomorphicfunction F in the idempotent representation, P (Z) = P1(β1)e + P2(β2)e

∗ andF (Z) = Φ(β1)e+ Ψ(β2)e

∗. Note that only a bicomplex holomorphic function canbe written this way, otherwise Φ and Ψ would be functions of both β1, β2. Thenthe equation P (D)F = 0 is equivalent to the system:

P1(∂β1)Φ(β1) = 0 ,

P2(∂β2)Ψ(β2) = 0 .

The result follows from Euler’s Principle for complex numbers applied to the poly-nomials P1 and P2. The varieties Vi for P are obtained from the correspondingzeroes of P1 and P2. �

This representation is consistent with the general Ehrenpreis’ FundamentalPrinciple. Indeed, a bicomplex holomorphic function can be interpreted as a mapwhose domain is R4. In order for the map to be bicomplex holomorphic, its compo-nents need to be holomorphic and satisfy the bicomplex Cauchy–Riemann system.With the request that such a function is in the kernel of an additional differentialequation, we now see that the variety that supports the function is discrete, andthat is reflected in the theorem we just proved.

The same result holds for a polynomial of a multi-complex variable P (Z)with Z ∈ BCn.


Theorem 5.2. Let P (Z) be a polynomial in BCn, where P (Z) =

n∑k=0

akZk. Then

any solution of the equation P (D)F = 0 where F is a bicomplex holomorphicfunction and D = ∂Z can be written as:

F (Z) =t∑

�=1

p�(Z)(eν�Z) ,

where the ν� are the zeroes of P , and p�(Z) are polynomials of degree m�−1, wherem� is the multiplicity of ν�.

The proof is similar to the proof for a simple bicomplex variable, it involvesreducing the bicomplex holomorphic function F to its 2n components in the idem-potent representation and we shall leave it to the reader.

We now extend the results above to the case of infinite-order differential

equations, by considering a series of a bicomplex variables such as f(Z) =

∞∑k=0

akZk

(analogous notations can be used in the case of BCn). As in the complex case weintroduce the notion of infraexponential type functions:

Definition 5.3. Let f be a function from BC2 to BC2, which is BC2-holomorphic.We say that f is of infraexponential type if for all ε > 0 there exists an Aε suchthat |f(Z)| ≤ Aεe

ε|Z|. We will denote the space of infraexponential type functionsby Exp0(BC2).

The space of infraexponential type functions is a proper subspace of the spaceof functions of exponential type defined by

Exp(BC2) := {f ∈ B(BC2)∣∣ |f(Z)| ≤ A exp(B|Z|), for someA,B > 0} .

One can easily prove the following lemma:

Lemma 5.4. If f is a bicomplex holomorphic function in BC2 then f ∈ Exp0(BC2)if and only if its components φ, ψ in the idempotent representation are in Exp0(C).Similarly, f ∈ Exp(BC2) if and only if its components in the idempotent represen-tation φ, ψ are in Exp(C).

Definition 5.5. Let P (Z) =

∞∑n=0

AnZn be a function from BC2 to BC2. We say that

P

(∂

∂Z

)is a differential operator of infinite order if and only if lim

n−→∞n n√|An| = 0.

We have the following two equivalence lemmas:

Lemma 5.6. The operator P (Z) =

∞∑n=0

AnZn is of infinite order if and only if its

components P1, P2 in the idempotent representation are of infinite order in C.

Proof. If P (Z) = P1(β1)e1 + P2(β2)e2 then An = ane1 + bne2 where an, bnare the coefficients of the series of P1, P2 respectively. We want to prove that

limn−→∞n n

√|An| = 0 if and only if lim

n−→∞n n√|an| = 0 and lim

n−→∞n n√|bn| = 0. The

direct implication is proven by a series of inequalities:

n n√|An| = n 2n

√|an|2 + |bn|2 ≤ n n

√|an|+ |bn| ≤ n n

√|an|+ n n

√|bn|.

The reciprocal is trivial and left to the reader. �

Lemma 5.7. P is of infinite order if and only if P ∈ Exp0(BC2).

Proof. This proof is a direct consequence of the previous two lemmas. �

Theorem 5.8. If P ∈ Exp0(BC2) then P

(∂

∂Z

)is a continuous surjective endo-

morphism of the space of bicomplex holomorphic functions.

Proof. The proof follows the known results in complex analysis. We have that Pis in Exp0(BC2) if and only if its components P1, P2 are of infinite order in C.From [11] we have that P1(∂β1), P2(∂β2) are continuous surjective endomorphismsin the spaces of holomorphic functions in the variables β1 and β2, respectively, andthe conclusion follows. �

Following [25], we obtain a series representation for solutions to infinite-orderdifferential equations in the bicomplex setting as follows:

Theorem 5.9. Let P ∈ Exp0(BC2) and let

V = {Z ∣∣P (Z) = 0} = {(αk,mk)∣∣ |α1| ≤ |α2| ≤ · · · },

where αk are bicomplex numbers with the respective multiplicities mk and |αk|denotes the Euclidean norm. Then there exists a sequence of indices k1 < · · · < kn

such that every entire bicomplex holomorphic solution of P

(∂

∂Z

)F = 0 can be

written as:

F (Z) =∑n≥1

⎛⎝ ∑kn≤k<kn+1

Pk(Z)eαkZ

⎞⎠ ,

where Pk is a polynomial of degree less than mk and the sequence of polynomialsPk satisfies the growth condition that allows the right-hand side of the sum aboveto converge.

Proof. The proof is a consequence of the equivalence lemmas above and the ratherintricate proof of the one variable complex case found for example in [25]. �


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[24] D.C. Struppa, Analytic functionals on the complex light cone, Advances in Hyper-complex Analysis Springer INdAM Series, Volume 1, 2013, pp. 119–124.

[25] D.C. Struppa, On the “grouping” phenomenon for holomorphic solutions of infiniteorder differential equations, Kyoto Surikaisekikenkyusho Kokyuroku, No. 1001, 1997,pp. 22–38.

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[27] D.C. Struppa, A. Vajiac, M. Vajiac, Remarks on Holomorphicity in three settings:Complex, Quaternionic, and Bicomplex, in Hypercomplex Analysis and Applications,Trends in Mathematics, Birkhauser Verlag, Basel, 2010, pp. 261–274.

[28] D.C. Struppa, A. Vajiac, M. Vajiac, Holomorphy in multicomplex spaces, SpectralTheory, Mathematical System Theory, Evolution Equations, Differential and Differ-ence Equations, Volume 221, Springer ISBN 9783034802970, 2012, pp. 617–634.

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[30] A. Vajiac, M. Vajiac, Multicomplex Hyperfunctions, Complex Variables and EllipticEquations, Complex Variables and Elliptic Equations, Volume 57, Issue 7-8, 2012,pp. 751–762.

Daniele C. Struppa, Adrian Vajiac and Mihaela B. VajiacSchmid College of Science and TechnologyChapman UniversityOrange 92866, CA, USAe-mail: [email protected]

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