function spaces of harmonic and analytic functions in infinitely … · 2. spaces of functions...
TRANSCRIPT
Function spaces of harmonic and analytic functions in infinitelymany variablesCitation for published version (APA):Martens, F. J. L. (1987). Function spaces of harmonic and analytic functions in infinitely many variables. (EUTreport. WSK, Dept. of Mathematics and Computing Science; Vol. 87-WSK-02). Technische UniversiteitEindhoven.
Document status and date:Published: 01/01/1987
Document Version:Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers)
Please check the document version of this publication:
• A submitted manuscript is the version of the article upon submission and before peer-review. There can beimportant differences between the submitted version and the official published version of record. Peopleinterested in the research are advised to contact the author for the final version of the publication, or visit theDOI to the publisher's website.• The final author version and the galley proof are versions of the publication after peer review.• The final published version features the final layout of the paper including the volume, issue and pagenumbers.Link to publication
General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.
• Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal.
If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, pleasefollow below link for the End User Agreement:www.tue.nl/taverne
Take down policyIf you believe that this document breaches copyright please contact us at:[email protected] details and we will investigate your claim.
Download date: 21. Jul. 2021
Eindhoven University of Technology Netherlands Department of Mathematics and Computing Science
Function spaces of harmonic and analytic functions in infintely many variables by EJ. L. Martens
AMS Subject Classifications: 81 B05, 46F05, 46F10
EUT Report 87-WSK-D2 ISSN 0167-9708 Coden: TEUEDE
Maart 1987
- 0 -
Contents
O. Introduction
1. Preliminaries
2. Spaces of functions related to spaces of harmonic functions
3. A space of functions on a real separable infinite dimensional
Hilbert space
4. A space of functions on the unit ball in IR£2
5. Special isomorphisms between M, the symmetric Fock space, the IN Bargmann space and (IR ,d~)
6. Characterization of M and related spaces
Acknowledgement
References
1
3
15
33
56
61
79
96
97
- 1 -
Introduction
In this university report we study spaces of functions on subsets of Hilbert
spaces. There are three main subjects in this paper.
The first subject is the theory of reproducing kernels. Most results on re
~roducing kernels which we apply, have already been stated in Aronszajns
paper [Ar]. We apply the theory of reproducing kernels to Hilbert spaces of
harmonic functions.
The second subject is the harmonic or spherical harmonic function. As a guide
and starting point we used the book Spherical Harmonics of Muller [MuJ.
The third subject is the representation of some special groups in the intro
duced spaces. The papers of KOno and Orihara [Ko] and [Or] served as a source
of inspiration.
The purposes of this report are the introduction of spaces of functions of
infinitely many variables, which are Hilbert spaces, analyticity spaces or
trajectory spaces. These spaces are characterized both functional analytical
ly and analytically.
Now we present a short summary of each section separately.
Section 1 deals with reproducing kernels, representations, analyticity on
locally convex vector spaces and Gegenbauer polynomials.
In section 2 we introduce a Hilbert space ~ of harmonic functions on lRq
and we study its reproducing kernel. We also consider the usual representa
tion of the orthogonal group on lRq in ~.
In section 3 we define a Hilbert space M of functions of an infinite number
of variables, which contains all spaces A~, q E IN. The fact that the set
of harmonic functions is dense in M is used to construct specific orthonormal
bases and to prove some general properties of the elements of M. In section 4 we define a Hilbert space N of functions on B the unit ball in
(U'2' We give results on this space N similar to the results of'the space M as defined in section 3.
In section 5 we consider isomorphisms between M and some well known spaces,
such as the Pock space, the Bargmann space and a space of square integrable
functions with respect to a Gaussian measure. For the latt.er type of space
we refer to [Ko] and [OrJ.
In the Bargmann space of infinite order the harmonic 90lynomiAls are dense
(Theorem 3.8). This is not true in a Bargmann space of finite order.
- 2 -
In section 6 characterizations of the space M and related spaces are present
ed. Now analyticity plays a more important role than harmoniticity. We have
two kinds of characterizations: Estimates with respect to reproducing kernels
or a combination of a growth condition and a (real) analyticity condition.
- 3 -
1. Preliminaries
In the first part of this section we present some general results on repro
ducing kernels and unitary representations of topological groups.
In the second part of this section we mention some recurrence relations of
Gegenbauer polynomials.
In the third part we introduce the notion of analyticity on a locally convex
topological vector space.
Finally we discuss an integral identity for c1-functions on a neighbourhood
of the unit sphere Sq-l in IRq.
Reproducing kernels
In this section F denotes a class of functions from a set D into ¢ which
establishes a complex Hilbert space with inner product (.,.) F and norm n. If",
Definition 1.1. The function K: D x D ~ t is called a reproducing kernel of
F iff
(a) V D: K{"Y) E F y£
f (y) •
We list some properties of reproducing kernels. These results can also be
found in [ArJ or [YosJ.
(1.1) For W c F let Span(W) denote the linear span of W, i.e.
Span(W) n
{ I j=l
The subspace Span({K(',y)
a.w. J J
Y ED}) is dense in F.
(1.2) Let the space F have a reproducing kernel K. Then for all x,y E D
II K(' ,y) I~ = KCy,y)
K(x,y) K
(1.3) The space F has at most one rcpr~lucing kernel.
- 4 -
(1.4) The space F has a reproducing kernel iff for each y E D the evaluation
functional f ~ fey), f E F, is continuous.
(l.S) Let F have a reproducing kernel K. Then we have
(1.6) Let Fl be a closed subspace of F, let F have a reproducing kernel K on
D x D and let P1
be the orthogonal projection from F onto Fl.
Then Fl has a reproducing kernel Kl with Kl (.,y) = Pi (K(.,y» and
Y E D, f E F •
(1.7) Let Fl and F2 be two mutually orthogonal closed subspaces of the function
space F with reproducing kernels Kl and K2 , respectively.
Then F admits a reproducing kernel K with K = Kl + K2 .
(1.8) Let F have a reproducing kernel K and let (~) be an orthonormal basis in F. . n n
The following equalities are valid:
I I(j)n (x) 12 = K(x,x) n
I ~n(Y)~n(x) = K(x,y) n
x,y ED.
(1.9) Let (~) be an orthonormal basis in F. n n
The space F admits a reproducing kernel iff for all x,y € D the series
L ~ (y)~ (x) is pointwise convergent. n n n
(1.10) Let DO denote a set and let K denote a function from DO x DO into ~ with
the property that for each n ~ 0, aj
( €, Yj ( D, 1 $ j
n
L ajatK(Y~'Yj) ~ 0 . j,t
$ n
- 5 -
Then there exists a vector space FO of functions on DO' which is a Hilbert
space and which admits K as a reproducing kernel. This space FO is uniquely
determined.
(1.11) The elements of a Hilbert space with reproducing kernel can be characterized
with the aid of the reproducing kernel:
(1. 12)
Theorem 1.2. Let F denote a Hilbert space of functions from a set 0 into ¢
with reproducing kernel K •. Let f denote a function from D into C.
Then f € F iff there exists c > 0 such that for each t E lN and all
ct. c Il, Yj 0 with
J
t I L ctl(Y j ) I j=l
Proof. *: Let t c lN, ct --- j straightforward estimate
t
I L ct/(Yj ) I j=1
1 S j s t
R. - ~ s c( I ~ctjK(Yk'Yj) } k, j=l
E (;, Yj € 0 where 1 :< j
t == I ( I u.K(.,y.) ,f) FI s
j=l J )
t S IIf IF II I K{.,yj)II
F
s t. Then we have the
-: Let W denote the Span({K(',y) lye D}). The linear functional m: W ~ C
is defined by
t t m( L ct
j K ( • , y j) ) = L ct.
j=l j=l J
R. c lN, ct. E 1:, Yj f. D where 1 s j s t. If J
t
L j=1
a.K(',y.) = a I
J )
then
R.
I j=l
a.f(y.) == a J )
- 6 -
because of (1.12), so the functional m is properly defined.
By assumption (1.12) and the properties of the re~roducing kernel of F it
follows that for all w E W
1m (w) I ~ ell w II •
The space W is dense in F, so by the Riesz' representation theorem there
exists 9 E F such that for all w E W
mew) == (w,g)F •
In particular for each y E D we have
fey) = m(K(·,y» (K(· ,y) ,g)
This implies f == g, whence f E F. rJ
Representations
Let G be a group with identity element e.
The Banach algebra of bounded operators on a Hilbert space X will be denoted
by L(X) and the identity on X by I or IX'
Definition 1.3. Let G be a topological group, let X be a complex Hilbert
space and let IT denote a mapping from G into L(X). The mapping IT is called a representation of the group G in X iff
(a)
(b)
(c)
V h G: TI(gh) = TI(g)IT(h) g, E
II(e) == I
V X= 9 + IT(g)x, 9 EGis continuous at e • XE
A representation IT is called unitary iff
(d) * -1 V G: [IT(g)] = IT(g ) . gE
Let F denote a Hilbert function space with reproducing kernel K: D x D ~~ <C
and let G denote a group of transformations on D. Then under certain condi
tions on K there exists a canonical unitary representation of G in F.
(1. 13)
(1. 14)
(1. 15)
( 1. 16)
- 7 -
Theorem 1.4~ Let G denote a topological group of transformations on O. Let F
have a reproducing kernel K with the properties:
(a) VR G V 0: K(Rx,Ry) = K(x,y) € x,y€
(b) Vy€O: The F-valued mapping R + K(-,Ry), REG is continuous at I D.
Then the mapping IT: G+ L(F) defined by
X E 0, f E F, REG
is a unitary representation of G in F.
Proof. Put W = Span({K(',y) lYE oJ).
For each REG the operator 'IT (R): W + W is defined by
t IT(R) ( I (),K(·,y.)) =
J ] Cl.K(-,Ry.),
J ] R, ?: 0, Cl. E e, y. € 0 •
J J
It is easy to see that
-1 [1T(R)f](x) = feR xl,
1T(RS)f = 1T(R)1T(S)f,
fEW
X E 0, f € W, REG
R,S E G, fEW
-1 ( 1T (R)f , g) F = (f, IT (R ) g) F' f,g E W, R E: G .
We only prove the last statement.
Let f,g E W with
k f = I Cl.K(-,a.) and
j=1 J J
Then for each R E: G
(1T(R)f,g)F ""
g
-1 (f,lT(R )g)F'
m = I B,Q,K(' ,b,Q,) .
1=1
- 8 -
The statements (1.14)-(1.16) imply for all REG, fEW:
Ihr (R) f If = II f If .
The set W is dense in F. Let Ti(R) denote the unique unitary extension of
1T (R) to F.
Let REG, let f E F and let XED.
Then we have
[1f (R) fJ (x) (n(R)f,K(o,x»F
-1 = (f,1T(R )K(-,x»F
-1 (f,K(o,R x»F
f(R- 1X) •
In the sequel we denote IT(R) by IT(R).
For all R,S E G it follows from (1.14) to (1.16)
II (RS) II(R)IT(S)
Let f E F.
Finally we want to prove that R + IT(R)f is continuous at 10 , E
Let £ > O. We take 9 E W such that II f - 9 II < '4 " Let
m
9 I j=l
CX,K( ,y,) J J
with cxj
E ~, CX, ~ 0, v, ED. J - J
Let Uj
be an open neighbourhood of 10 such that for all R E Uj
m
Put U = n Uj
< Then U is an open neighbourhood of 1D" j=l
For R E U we have
m
~ t + I I CX]' I II K ( • , RYJ') - K ( • , yJ,) n F+ :i < £ " 0
j=l
- 9 -
Let V be a subspace of the Hilbert space X and let A E L(X) be such that
A (V) c V.
This subspace V is called A-invariant.
The restriction of A to V will be denoted by Alv' It is clear that Alv E L(V). * If also A (V) c V then V is s.aid to reduce A. If U is a unitary operator then
each V-invariant subspace is also reducing.
Definition 1.5. Let n denote a unitary representation of a topological group
G in the Hilbert space X. The representation n is called irreducible iff each closed subspace V
of X, which is TI(g)-invariant for all g E G, equals {a} or X.
Lemma 1.6. Schur's lemma.
Let IT be a irreducible unitary representation of a topological group G in the Hilbert space X. Let A E L(X) be such that for all g E G the
operators A and neg) commute.
Then there exists A E ~ such that
A = AI .
• See [so], I, Satz 2.3. ---
Theorem 1.7. Gurevi~.
Let IT be a unitary representation of a compact topological group G in
the separable Hilbert space X.
Then there exist finite dimensional invariant subsoaces X , n E IN, of '-n X such that
(a)
(b)
X 1 X n m
X = (9 X n
n
n :F m
(c) g + IT (g) Ix ' g E G, is an irreducible unitary representation of G n
in X for all n E IN . n
Proof. See [so], lY, Satz 1.2. o
(1.17)
( 1.18)
(1.19)
- 10 -
Gegenbauer polynomials
For q 2 2 and n ;::: ° we define the function pq
: :IF -+ lR by n
Lemma 1.8. Let q ~ 2.
Then the following statements are valid.
(a) pq is a polynomial of degree n . n
(b) ° I (c) 1 , n E IN •
Proof. See [MuJ, p. 16-18.
n :f m
In [MOS], § 5.3 the Gegenbauer polynomials c A of degree n, n ~ 0, and order n
o
A, A ;::: -~, are introduced as special cases of Jacobi polynomials. These poly-
nomials satisfy
1
f c A (t) c A (t) (1 _ t2)A-~dt = ° , n m n ;. m
-1
"" I CA(t)rn 1
= , n=O n
(1 - 2tr + r2)A A ;. 0 I
t E [-l,lJ, Irl < 1
"" I CO(t)rn -log (1 -
2 2tr + r ) n=O n
Lemma 1. 9. F ix q ;:: 3.
Then
pq = r(n + l)r(q - 2) c~q-l n r{n + q - 2) n n ~ 0
p2 =:; ~ cO n 2 n' n 2: 1
1
(1. 20)
(1.21)
- 11 -
Proof. 14' rom Lemma 1. 8 (b) and re lation ( 1. 17) we obtain a q E lR such that n
~rom Lemma 1.S(c) and relation (1.18) it follows
(aq}-l = c~q-l(l) = r(n + q - 2) n n r(n+ 1) (q-2)'
The proof for p2 runs similarly. n
We present two recurrence relations for the pq,s. By pq we mean (Pq) '. n n n
Lemma 1.10. Let q ~ 2 and n ~ O. Define P~1 (t) = O.
Then we have
(2n + q - 2)t pq(t) n (n + q - 2)P~+1 (t) + n P~-l (t)
and
(2n + q - 2)P~(t) = (n(~; ~)2) P~+l (t) - (n +: _ 3) P~-l (t)
Proof. From [MOS], § 5.3, we have
2(n + A)tC~(t) = (n + 1)C~+1 (t) + (n + 2A - 1)C~_1 (t)
and
With the aid of Lemma 1.8 we get the desired results.
Lemma 1.11. Let (t) be a Cauchy sequence in IR with limit t. q q
Then for all n ~ 0
Proof. Since pq - 1 we have lim P6(t ) = 1. 0 q-l-<X> q
Assume for 0 ::; j ::; n
lim P~(t ) t j
q-)<X) ] q
Because of Lemma 1.10
o
t E IR
o
- 12 -
pq 1 (t ) n+ q
(2n + q - 2) t pCJ (t ) (n + q - 2) q n q
n pq l{t ) (n + q - 2) n- q
whence lim pq l{t ) n+ q
q~
n+l = t
1<' or q,m € IN and n ::>: m define the function Aq
+1
: [-1,1] -l> m. by n,m
(1.22) Aq+1 (t)
n,m
Lemma 1.12. Let q,m,n1
,n2
t IN with nl
,n2
~ m.
Then
(a)
+ 1
(b) Aq+1 (t)
rem + ( _____ ) 1:1(1 _ t2)m/2 •
m,m r (m + !)!;
Proof.
(a) See [MU], Lemmas 14 and 15 and pages 3 and 4.
(b) Because of Lemma 1.10 the coefficient of t m in pq+1 is positive. m
By (1.22) there exists a > 0 such that
Aq +1 (t) m,m Since
-1
we have
Analyticity
q + 1 rem + -2-) 1:1 ( )
r (m + !) !;/r (m +
The notion of analyticity will be generalized.
o
- 13 -
fEfini tion 1. 13. Let W denote an open subset of «:q.
The function f: W -+ C is called analytic on W iff for all w € W there
exists an open neighbourhood N of w such that for all ZEN w w
f (z)
where the series is absolutely convergent.
k (z - w ) q
q q
fEfinition 1. 14. Let V denote a complex (real) locally convex topological
vector space and let W denote an open subset of V' •
The function f: W -+ it is called analytic on W iff f is continuous on W
and for all w ( Wand v (' '/ there exist an open neighbourhood M of . w,v o in it such that A -+ f(w + AV) is analytic on M (is extendible to an
w,v analytic function on M ) .
w,v
If f is analytic on an open subset W in cq because of definition 1.13, then
certainly f is analytic because of definition 1.14.
Theorem 1.15. Hartogs theorem.
If u is a complex valued function defined on an open set ~ c Cq
and u
is analytic in each variable z. when the other variables are given arJ
bitrary values, then u is analytic in Q.
Proof. See [HoJ, theorem 2.2.8.
An integral identity
[J
Let sq-l and sq denote the unit sphere and the o~en unit ball in mq , respec
tively. q-l The usual surface measure on S will b~denoted by dOq_
1 .
1~~~~j~j~ Fo~ a C'-function on a neighbourhood of sq the normal d~rivative in a point
I; E sq"1 wi 11 be denoted by
d f (0 v
- 14 -
Theorem 1.16. Let f and 9 be continuously differentiable functions on a
neighbourhood of the closed unit ball Bq
• Let j c IN with 1 ~ j ~ q.
Then we have the following identity
a [ax. fJ(I;)g J
f ( ~) 9 J ( ~) dO' 1 (~) + ax. q-J
+ J l;.[a f(l;)g J v
+ f ( t;) a 9 ( t;) J do 1 (l;) + v q-
Proof. Put
sq-1
+ (q - 1) J t;.f(t.:)g{t;)dO 1(1;;) • J q-
q-l S
[~. fJ( 9 J
() f (I;) [-" - 9 J ( l;) do 1 ( l;) • ox. q-
]
Straightforward calculation yields
M = I [d -ax. (f. g) ] ( 1;) d a q-l ( t;) J q-l
S
I k=l
q-l S
B
f () [3 --3 - -a -(f.g) ] (x) xkdx ~ x.
q J
+ q I a -
[ax. (f.g) J(x)dx = J
(Gauss theorem)
J q-l
S
B
far () -~-a -( f . g) (x) x. Jdx +
x. ~ k q J
f a -(q-l) [ax.(f.g)](X)dX
J
s·[a f(l;)g{t;) + f(l;)a g(t;)JdO I(!;) + J v v q-
+ (q - 1) J l;jf(t;)9(I;)doq
_I
(1;) •
sq-1
o
- 15 -
2. Spaces of functions related to spaces of harmonic functions
In this section we introduce spaces of fUnctions which are restrictions of
harmonic functions to sq-l or are harmonic functions on IRq. Most results
are extensions of results in [Mu] and [EG].
We use the following notations where we take q £ lN, q > 1, fixed.
q-1 S (a,r)
dx
dO' 1 q-
2rrq / 2 0' q-l r (s.)
2
f.:, r a2
q 2 j=l ox.
J
a r = x. q,v l. j=1
Harm(W)
( ... ) II • II
a ax.
l.
, the sphere with centre a and radius r in IRq
, the unit sphere in IRq
, the open ball with centre a and radius r in
IRq
the open unit ball in IRq
the usual Lebesgue measure in IRq
, the usual (q-1)-dimensional surface measure q-l
on S
Note that dx
q-l , the total surface measure of S
, the q-dimensional Laplacian
, the q-dimensional normal derivative
, the space of complex valued functions f
which are harmonic on a neighbourhood of each
point of W c IRq
Note that f € Harm (sq) implies
f E Harm(Sq(O,l + Ell for some E > O.
, the compact group of orthogonal transforma
tions from IR q onto IR q
, subgroup of O(IRq ) consisting of all rota
tions R(det(R} = 1)
, the inner product on IRq
, the norm on IRq •
(2. 1)
- 16 -
q-1 q-l Consider the Hilbert space L
2(S ) = L
2(S ,dO
q_
t) of square integrable
q-l functions on S with the natural inner product
f ( ~) g ( ~) do 1 ( t;;) , q-
We denote the corresponding norm by II. II 1 • q sq- q-1
For each R E O(IR ) we define the operator LR
: L2
(S )
Then we have:
- LRS LRLS
- LIf = f ,
q R, S € 0 (IR ). q-l
f € L2
(S ).
The mapping R + LRf is continuous
- The operator (LR
)* is equal to L T
the surface measure do 1 q-R
q-l for each f E L
2(S ).
because of the rotational invariance of
q q-1 So R + LR is a unitary representation of O(IR ) in L
2(S ).
Since O(IRq ) is a compact group, theorem 1.7 yields the existence of mutual
ly orthogonal finite dimensional subspaces Vq n ~ 0, of L (sq-1) such that n' 2
- L2
(Sq-1) = $ Vq • n=O n
The subspaces V~, n ~ 0 remain invariant under the operators LR
, R € O(IRq),
and the representation R + LRI is irreducible. Vq
n In [Vi], Chapter IX, § 3 the following has been shown:
The subspaces Vq can be selected in such a way that for each f E Vq
there n n
exists a unique harmonic homogeneous polynomial p of degree n in q variables
(i.e. p(AX) = Anp(X) and [6~](X) = 0, A € IR, x E IRq) such that
f(~) = p(~) ,
This leads to the following definition:
Definition 2.1. Fix q € IN and n ~ O.
Let ~ denote the space of harmonic homogeneous polynomials of degree n n
in q variables.
Put dq dim(~) dim(Vq ). n n n
- 17 -
Lemma 2.2. Let q ~ 2.
The numbers dq
can be explicitly calculated. We have n
dq
= (2n + q - 2)f(n + q - 2)/(r(n + l)r(q - 1» n
dq
1 o .
Proof. See [Mu], pp. 3 and 4.
n ~ 1
Since Vq is finite dimensional, it follows from remark (1.9) that Vq
has a n n reproducing kernel. The following lemma yields an explicit formula for this
reproducing kernel.
Lemma 2.3. Let {eq . I 1 ~ j ~ dq } be any orthonormal basis of ~. n,] n n
Then we have
dq n----
I j=l
Here the polynomial pq is defined by (1.17). n
Proof. See [Mu], theorem 2.
Corollary 2.4. Let f E Vq•
n Then
If(e;) I
Proof. See property (1.5).
q-l ~,n E S .
Let Dq denote the orthogonal prOjection from L (sq-t) onto liq • n 2 n
Corollary 2.5. Let f E L2
(Sq-l).
For each n E sq-l we have the equality
o
o
o
- 18 -
Proof. See property (1.6). o
In the following lemma we present the generating function of the polynomials
pq. n
Lemma 2.6. Let t,t € IR with It I < 1 and It I ~ 1.
Then we have
L n=O
For fixed t the series is uniformly convergent in t, ITI ~ 1.
Proof. From Lemmas 1.9 and 2.2 it follows
q q (2n + q - 2) c~q-l d·P = -n n (q - 2) n
Relation (1.17) gives the wanted result. IJ
Corollary 2.7. Let t f IR with It I < 1.
Then we have
L n=O
1 +t
(1 - t) q-1
Proof. Take t 1 in Lemma 2.6. o
Now we are in a position to link to each f € L (sq-l) a harmonic function ~f 2
on a ball Bq(O,r) where r ~ 1 depends on f.
q-l Definition 2.8. Let f E L2
(S ). Then
Let
r = sup{p ~ 1
The function Hqf: Bq(O,r) ~ ~ is defined by
(2.2)
- 19 -
We remark that the series in (2.2) is absolutely and uniformly convergent on
each ball Bq(O,p) with p < r. This is a consequence of the definition of r
and the corollaries 2.4 and 2.7.
q-1 Theorem 2.9. Let f E L2
(S ) and let r be as in definition 2.8.
(a) The function Hqf is harmonic on Bq(O,r) .
(b) If there exist 5 > 1 and a harmonic function g on Bq(O,s) such that
g(~) f(~) for ~ E sq-l, then
(i) r ~ 5
(ii) [Hqf] (x) = 9 (x) , q X E B (0,5).
(c) For y € Bq the following integral representation holds:
[Hqf](y) = ~ J q-l q-1
S
1 - II Y Ir f ( ~) do ( ~ ) • "~ _ y f q-l
Proof. (a) and (b): See [Gr], Lemma 2.1.
(c): Take y E Bq, Y ~ O.
Then we have
"" dq
J :: I ~"Ylr f(~)P~(II~ II .~)dOq_l (~)
n=O q-1 S q-l
(Corollary 2.5)
1
J 1 - Ill' Ir f (~) do 1 (~) =
0 q-l (1 + lIy It - 2(y.E;:»Q/2 Q-q-l
S (Lemma 2.6)
-=---.::.....~ f(~)da 1 (~) . II ~ - y IF q-
=
Moreover
Let n ~ 1.
[Iflf] (x)
[Hqf] (0)
So we see that for
Let f € Vq•
n
= II x IPf (II: Ir
= 0 .
all 11 € S q-l
- 20 -
f(~)dcr 1 (~) . q-
Then
We remark that the mapping Ifl: L2
(sq-l) + Harm(Bq ) is injective. - -1
Next we consider the spaces Harm (Bq ), Ifl(L2
(Sq » and Harm(Bq). The first
and the second are contained in the latter. So we get
o
This triple of spaces fits in a functional analytic setting which we describe
now.
In [EG] the analyticity space S B and the trajectory space T Bare intro-Y, Y,
duced for Y a Hilbert space and B a self-adjoint operator.
Here
I tB {f € Y 3
t>0: f E D(e )}
and
They constitute the triple
As we shall see
for a suitable self-adjoint operator A.
So all functional analytic results of [EG] apply here.
- 21 -
Definition 2.10. The self-adjoint operator A in L (sq-l) is defined by 2
Af I f E D(A)
n=O
The operator A has the following properties:
2 ~ (2.3) - The operator A is equal to -~(q - 1)1 + {~(q - 1) I + ~LB} where ~B de-
q-l notes the Laplace-Beltrami operator of the sphere S •
(2.4) - For each f E D(A) with ~f E Harm (Bq )
Af (i;) [d ~fJ(i;) q,v
because a h = nh for all h E Hq. q,v n
(2.5) - Fix t > O.
For all f € D(e(log t)A)
Remark:
D(e(log t)A) =
From (2.5) it follows
The Space T 1 can be represented by Harm(F3q ). To this en.d we extend q-L2 (S ),A
the operator Hq : L2
(Sq-l) ~ Harm(Bq )
tension will be denoted by gq. to an operator on T The ex-
L2
(sq-l),A"
For each F € T we define q-1 < L2 (5 ) ,A
x [F (-log II x ID] (9)
[~FJ<O) = [F(1)](0) , 1 > 0 •
We observe that F (-log II x ID E S q-1 • L2 (S ) ,A
- 22 -
The function gqF is harmonic on Bq
•
Further gq is a linear bijection from T 1 onto Harm (Bq
) • Also this q-
is a consequence of (2.5). L2 (S ),A
If moreover F E L2
(sq-l) (i.e. F(t) e-tAf for some f E L2
(Sq-l» then
'if1F Ff!f.
Next we introduce some operators in L (sq-l) which will be needed later on. 2
Definition 2.11. We define the operators Pk
and ~, 1 ~ k ~ q, in L2 (Sq-l)
by
Lemma 2.12. Let f E ~ and let k E 1N with 1 ~ k $ q. m
The following relations hold true:
~~f = (2m + q - 2)Dq
(0 f) J\. m-l -X
Proof. Fix ~ E sq-l.
Applying corollary 2.5 and lemma 1.10 we get
[~fJ (t;)
o q-l f
q-l S
q ~,.P (n.t;)f(n)do len)
J\. m q-
l:" [(m +q - 2) • q "k - (m + 1) P m+l (n. t;)
m·q fen) (m +q -3) Pm- 1 (n.1;;) J(2m +q -2) dOq _ 1 (n)
~(m +q -2) pq ( •• ~) _ m dY
k (m + 1) m+l (m +q
p q ( •• t;) ] (11) f ( n) do 1 (n) . 3) m-l 2m +q -2 q-
Next we want to apply the integral identity of theorem 1.15.
- 23 -
To this end we remark that
[ a (JIlf) ] (nl q,v
= m (EfIf) (n) = mf (n)
and also that
[-f-(Hqf) ](n) = [Pkf] (n) . Yk
So we get
cr q-l f
nkf(n) a [m+q-2 pq (e.l;) __ m_ pq (-.l;)](n) dcr len) + q,v m + 1 m+l m+q-3 m-l (2m+q-2) q-
q-l S
dq
+_m_ J [(m+ q -2) Pmq+1(n.l;) - m pq (n.l;)]' cr 1 (m+l) (m+q-3) m-l q- q-l
S
Now we have
dq
m J [(m+q -2) oq Ll =G m :; 1 Pm+ 1 (n.l;)
q q-l
nk
f (n) m eq (m+q-3) Pm-l (n.l;)](n.~) (2m+q-2) dcrq _1 (n) •
S
Applying Lemma 1.10 twice yields
dq
=~ J q-l q-l
S
whence
:iI: ..... em + q - n, (2m +q -2) (m +1)
dq
m f pq (T'!. t;;) [ ..... ('''f(n) + (2m.+q) 11kf (nl Jdcrq
_1
(11) + cr m+1 k . q-l
sq-l
(2.6)
- 24 -
dq
m m (2m+q-2)(m+q-3) (1q_l J P~-l (n.~) [-(Pkf) (n) +2nk
f (n) ]dO"q_l (n).
sq-l
S' P f E: Vq and ~ + q - 2 dq l.nce k m-l (2Q, + q - 2) (9, + 1 ) R,
= 1 dq R, ~ 0 we get
(2Q. +q) H1 '
+ ___ 1 ___ Pkf) --(-2m--.-.:;;=-q----:- D~_l (~f)
Application of the projection D~_1 on both sides of the latter equality yields
If we substitute this in (2.6) we arrive at the other wanted relation. 0
In the final part of this section we discuss another Hilbert space which con
tains all ~ as mutually orthogonal subspaces. Let n
with natural inner product
( 21T) f exp (-II x Ir /2) f (x) 9 dx,
IRq
and with corresponding norm " • " M2 (IRq)
2 13 Let ~ denote the algebraic direct sum of the Hq,s, i.e • .;;;....;c.~~..;;..;;;;...;...;..;;.-"'-:....;;;..;~- R, n
each f E ~ is of the form f = I n=O
f with R, ~ 0, f E: Hq, 1 ~ n ~ £. n n n
Observe that Hq is the space of all harmonic polynomials in q variables.
Let J..fI denote the closure of Hq in M2 (IRq) •
Our first aim is a characterization of the elements of ~.
Lemma 2.14. Let 9 E Vq and h E Vq
• n m
Then we have the following equality
2 (n+m) /2. r (n +~ +q)
2.1Tq / 2 9 (~) h (~) dO" 1 (f,;) q-
- 25 -
Pro.cE., We. have,
1 . J exp(-lIxlfI2) (Hqg) (x) UfIh) (x)dx (211') q/2
IRq
OC>
= I' J (r2) n+m+q-l q/2 exp - :2 r
(2'11') . o
I g(~)h(~)dcrq_l (~)dr sq-l
2 (n+m)/2
q/2 2.'11'
<X>
I () (n+m+q-2) /2 f exp -s s
o 1 sq-
2(n+m)/2. r (n +~ +q)
= --------~--~~-2.n-
g/2 J g(~)h(~)dcr 1(~) q-
q-l S
Corollary 2.15. Let n ~ m. <X>
Then ~ ~ Hq in ~. Moreover ~fI = ~ Hq
• n m n=O n
Proof. Vq 1 Vq in L
2(SQ-l).
-- n m The second statement is trivial.
Corollary 2.16. Let n ~ O. Let {eq . I 1 ~ j ~ dq
} be an orthonormal basis n,J n
in Vq. Then
n
is an orthonormal basis in Hq•
n
Co.rollary 2. 17. Let n ~ 1.
The subspace Hq has a reproducing kernel Kq
with n n
o
x .".. 0 .". y
x "" 0 or·y = 0
o
o
(2.7)
- 26 -
Proof. Let {eq
. I 1 ~ j ~ dq
} be an orthonormal basis of ~. Fix x,y E IRq n,] n n
with x f: 0 f: y.
Because of corollary 2.16 and property 1.8
In case y = 0 or x
q/2 dq
2.n IIx If I! If _n_ pq(~ 2nr(n+!l) y O'q_l nUx!!
2
(Lemma 2.3)
o we have Kq(x,y) n
o.
Of course the kernel Kci of Hci satisfies
y • lIy I~ ;:
Because of property (1.5) we know that for each f E Hq n
X E
Lemma 2 1 . Let n ~ 0 and let x E IRq. Then
1 we have
o
(2.8)
Because q ~ 2 we
a. + 9: -2a. + q
So we get
Kq(x,x) n
Further we have
q K1 (x,x)
and q
KO (x,x)
=
- 27 -
rei) (2n +q -2) r(n +q -2)
2nr(n + !) r(n + t) r(kI - 1)
(2n + q - 2) (n - 2 + q - 1)
(2n - 2 + q) (2 (n - 2) + q)
have for each (l ~ 0
1 s....::..1:. :5 . q
:5 (~)n-1 IIx Ifn
q n!
:5 II x It (because dq 1
q)
1 :5 .9....:....l -1 ( q )
Corollary 2.19. Let f € ~ and x € IRq. Then n
n-1
I f (x) I q -:<;; ( q
1) -2- II x Ifll f II
rnr M2
(IRq)
Proof. From inequality (2.7).
(q - 1) II x Ifn
q r(n + 1) •
Let Qq denote the orthogonal projection from ~ onto ~. n n
Theorem 2.20. Let f € ~.
Then f is a harmonic function on IRq which satisfies
I f (x) I
Proof. Pix x E IRq. For all g € ftt· we have
n-1
o
o
I I n=O
I n=O
I [Q~g] (x) I ~J"J~;,l) ";~'~~!k~:r ... q) <
- 28 -
(Corollary 2. 19)
()O
s ( I n=O
/ q exp(~ q - 1 q
So the linear functional t : ~ ~ € defined by x
t (g) = x
is continuous.
00
I [Q~g] (x) I
n=O
For all 9 E Hq we have t (g) = g(x). x
m For m € IN put f = I [Qqf].
m n=O n Then f + f, m + "" in norm and f + ~ t (f), m + "", uniformly on compacta in
m m x x IRq. Since the f 's are harmonic, the function f is harmonic. It is clear
m that f satisfies the inequality (2.8). 0
Next we prove the converse of theorem 2.20. We need an auxiliary result.
Lemma 2.21. Let r > 1. Let f E Harm(Bq(O,r» and 9 E Harm(Bq ). For each P,
1 s p < r
Proof. We have
=
f(~)g(i;)da 1 (~) = q-
f(pE;;> [e-(log p)Ag](t:;)da 1(~) = q-
[e-(log p)Af(p.)J(t:;)g(~)da 1(t:;) q- f (~) 9 (E;) do 1 (~) . 0
q-
- 29 -
Proof. From theorem 2.20 it follows that if! c M2 (mq ) n Harm(mq). Let q q .a q-l
f E M2 (m ) n Harm(m ) and assume that f .l M-. For all 11 E Sand n ~ 0
We have
1 = ---=-........,.-(21T)q/2
J exp(-lIx If/2)f(X)K~(X,n)dX =
mq
o
From Lemma 2.21 it follows that
So for all 11 € sq-l and n ~ 0
a
[
"" 2 r n+q-l
exp(- 2) r
n r
a q-l
f q-l
S
So fl q-1 .l L2 (Sq-l). Hence for each n E sq-l fen) = O. Since f is harmonic, S
we conclude f = O.
$.E;tna~ks.
- From Theorem 2.20 it follows that for all x E IRq the linear !unctionals
f ~ f(x)t f E ~ are continuous. So the space ~ has a reprOducirtgkernel 00
e n=O
Hq it follows from n~.~n<.~+·" that n
00
L n=O
o
We have for f E rfI n
- 30 -
(lemma 2.14) •
So it follows that Bq is a compact injective operator from MJ. into L2
(Sq-l).
Next we define a unitary representation of O(IRq ) in MJ.. For all f E ~~ and
R E O(IRq
) the function f(RT .) ( ~f1. The following definition makes sense:
Definition 2.23. The representation rrq : O(IRq
) + L(~~) is defined by
It is easy to see that nq is a unitary representation of O(IRq
) in ~.
Each finite dimensional subspace rfI is invariant under the operators rrq(R) , n
R E O(IRq).
The representation R + rrq(R) IrfI' R E O(IRq),iS irreducible, because rrq(R) :=::
n = HqL B
q and because the representation R + L I ' R E O(IR
q) is irreducible.
R 00 R~
r1oreover MJ. = $ rfI. See theorem 1.6 (Gurevic). n n=O n
Finally we give another description of the inner product (-,.) restrict-ed to Hq. M2 (IRq)
Definition 2.24. Let p be a polynomial in q variables. a Let p(3) denote the differential operator p(--- , .•• , aX
1
Lemma 2.25. Let f,g E rfI.
Then (f,g) :=:: [f(3)9J (0) M
2(IRq )
d ax) . q
- 31 -
prodf. If f € ~ and 9 € ~ with n# m, then (f,g) q = 0 and n m [f(a)g](O) = O. M2 (m )
So we restrict ourselves to f,g €
Because of Lemma 2.14 we have
/fl. n
= -----=-0' q-1
f(t;)g(E;)dO' 1 (t;) q-
The homogeneous polynomial f is a linear combination of monomials of the form n II xk . with 1 S k.~ q.
j=1 J J q-l Lete - 1 on S • Put
2nr (n + 9.) 1
f n
a = II t;k. g dO' 1 (t;) r (9.) a
q-l q-
2 q-1 J S
Then we have
2nr (n + ~) 1
n a = [ IT ~ e,g]q
f(9.) tJ j=1 2
q-l j
Because ~ is self-adjoint and because of Lemma 2.12 n
.
2nr(n + 9.) n-1 a = ___ --=;:..2_ a [IT ~ e, D~+l (~ g) + D
q 1 (~ g) ]
r(9.) q-l j=1 j n n- n q 2
n-l Remembering that II ~.e ~ V~+1 and applying Lemma 2.12 again we ~et
j=1 J
n-l f- [IT ~.e, q-l 1 J
2n- 1r(n - 1 + 9.) n-l ______ 2 __ 1_ [ IT
°q_l j=1
Since Pk 9 € H~_l we can repeat this argument n-l times and finally we get n
- 32 -
1 n
(l = -=-----r e , a q-l
IT Pk
g] j=l j q
1 =--a q-l
f q-l
S
n ( IT ~ g) (~)da 1 (~)
aX.. q-j=l k J
n [IT -f- g] (0) j=l Xk
j
Since f is a linear combination of monomials we get
(f,g) q = [f(Cl)g](O) . M2 (IR )
o
- 33 -
3. A space of functions on a real separable infinite dimensional Hilbert space
In the first part of this section we construct a complex Hilbert space of
functions of infinitely many real variables with the aid of the Hilbert
spaces of harmonic functions, introduced in the previous section.
In the second part of this section a unitary representation of the full or
thogonal group on IRt2 in this Hilbert space will be introduced.
We use the following conventions and notations:
(3.1) IRt2 denotes the real Hilbert space of square summable sequences with inner
product and norm denoted by ( ••• ) and II • II respectively.
For a,b,c,d E IRt2 we define
(a + ib. c + id) = (a.c) + (b.d) + i[ (b.c) - (a.d)] .
(3.2) (ej)j denotes the standard orthonormal basis of IRt2 ; i.e. e j
(3.3) IRtc denotes the subspace of IRt2 consisting of all finite sequences x i.e,
only a finite number of entries of x are non zero.
(3.4) The space IRq is embedded in IRt2 identifying (x1' ... ,xq ) and ~ x,e. E IRt2' j=1 J J
Thus we get for example
IRq = IRt and so on . c
(3.5) O(IRt2
) denotes the group of all orthogonal operators (i.e. bijective isome
tries) from IRt2 onto IRt2
.
(3.6) o ( IR t ) == {V E 0 ( IR t2
) I V ( IR t ) = IR t } • c c c
(3.7) Eq denotes the orthogonal prOjection from IRt2 onto IRq.
(3.8) The group O(IRq
) is embedded in O(IRt2
) identifying R E O(IRq
) and the ortho
gonal operator RE + (I - E ) E O(IRt2).
00 q q
Remark: u O(IRq ) is a proper subset of O(IRt ). c
q=l Example: Define U E O{IRt
2) by
00
Then U t u q=l
n E IN •
O(IR q ) and U E O(IRt ). c
(3.9)
(3.10)
(3. 11)
(3.12)
- 34 -
We start with the definition of an inner product space in which all spaces
~ of the previous section can be embedded.
Definition 3.1. The linear space F(IRt2) consists of all functions f: IRt2 ~ ~
with the following properties:
(a) f is continuous on IR!2'
(b)
(c)
3 : f = foE p>O p
I e -II x If 121 f (x) 12dx < CD for p IS IN such
IRP
In F(IR!2) an inner product is defined by
(f,g) F(IRt ) 2
that f
The corresponding norm will be denoted by II. I'F (IR t ). 2 (Remark:
CD
f e-u2/2
dU = (2n)~) .
_00
Definition 3.2. Let q € IN.
The mapping embq
: ~ E F(IR!2) is defined by
embqf = foE q
foE • P
If no confusion arises, we identify f € ~ and embqf € F(IRt2
) and we denote
embqf by f.
So ~ will be considered as a proper subspace of F(IRt2). With these conven
tions we have the following relations:
(f,g) F(IR! ) 2
(f,g) q M2 (IR )
and
tfI c Hq+P c F(IRR.2
) n n
for q,p E IN
for q,p € IN
II f I'FCIR! ) = 2
f,g € ~
- 35 -
Lemma 3.3. Let m,n,p,q E IN with m F n.
Then HP 1. Hq in F {IR 9..2
} • n m
Proof. Let r max{p,q}.
By (3.10) HP c Hr and ~ c Hr ,
n n m m
Since Hr
1. Hr
in ~f, by (3.9) we conclude HP .1 ~ in F(IR9..2).
n m n m
In the sequel the following subspaces of F(IR9..2
) will be of importance.
Let H denote the subspace n
Let H denote the subspace ~ ~. q=l
00
00
u q=l
o
We recall that ~ Span ( u n=O
~) t n
i.e. the space of all harmonic polynomials
in q variables.
Observe that H is the direct sum of the H '5, so H is the space of all harn
monic polynomials of an arbitrary number of variables. For the spaces of de-
finition 3.4 we construct a Hilbert completion with respect to II. If(IRt )' 2
To this end some auxiliary results are needed,
Lemma 3 ,Let f E H and let x E IRt2' Then
I f (x) I IIx Ir ~ exp (-2-) II f If (IR9. )
2
Proof, Since f E H I there exists apE IN such that f € ~ c ~f! for all q ? p,
From theorem 2.20 it follows
I f (x) I So
I f (x) I
If (E x) I p
II xli 2
exp (-2-)11 fll F(IRt ) • 2
o
Theorem 3.6. Let (fk)k be a Cauchy sequence in H. (a) Then (fk(x»k is a convergent sequence in ~ for all x € IR~2.
(b) Moreover the function f: IR~2 ~ ~ defined by
f (x) = lim fk (x) ,
k-"
satisfies the following conditions:
( -I) If( ) I (II x If)l' Ilf IL ... x ~ exp -2- ~m k 't- (IR ~ ) ,
and k+<x> 2
(ii) f is continuous on IR~2.
Proof. (a): Fix x E IR~2.
By lemma 3.5
So (fk(x»k is a convergent sequence.
(b) (i): Observe that
If(x) I = lim Ifk(x) I ~ k-..
IIx If , :s; exp (-2-) hm II fk IhIR 11.) •
k+<x> 2
(ii): Fix x € IR11.2 and e: > O.
Let k > 0 such that for all R, ~ k
2 II f f IL ( (II x II + 1) ) ~
k - R, 'f (IR R, ) :s; exp - 2 b • 2
Let 1 > 0 > 0 be such that Ifk(x) - fk(y) I <: f for all y ( IRt2 with
IIx-yll<:o.
Then for all t <:: k and all y wi th II x - y II <: 0
:::; ~ ex (II x If _ 01 x II + 1) 2) + .£ + ~ (hl_ <II x II + 1) 2, <: ~ 6 P 2 2 6 6 exp 2 2 I - 2 •
- 37 -
So for all y € mt2
with II x - y II < 0
If(x) - fey) I < e: • o
Defini tion 3.7. Let M denote the Hilbert completion with respect to II • II F (:IR R,2)
of H which consists of all functions on mt2
which are pointwise
limits of Cauchy sequences in H. The corresponding inner product and norm in M will again be denoted by
( • ) fA and II • 1M, Let M denote the closure of H in M.
n n
Each element of M is the pointwise limit of a sequence of harmonic functions
each of an arbitrary but finite number of variables.
Since H L H , n ~ m, and H = n m u nEIN
H n
it follows that M n
L M , n ~ m and that m
M =: q) M. n=O n
The space ~~ is a closed subspace of M because of (3.9). Since M consists n
of homogeneous functions, we have M n ~ = ~. n n
(3.13) Let pq, P and pq denote the orthogonal projections from M onto ~, M and n n n n
W respectively.
Lemma 3.8. Let n € IN.
(a) pq + P , q + 00, strongly . n n
(b) pq + I, q + 00, strongly .
Proof. (a): Take f E M. Let e: > O.
We choose g € H such that IIf - gil < t Take q E IN with g E Hq.
Then for p ~ q
IlPPf - P f II ~ IIPP(f -g)IIM
+ IIPPg - P gllM
+ lip {g - f)IIM n n M n n n n
<
(b): Similar to (a). o
Lemma 3.9. Let x € IR12 .
(a) For all f € M
I IIx If If(x) ~ exp (-2-l II filM
(b) For all f € M n
- 38 -
Proof. (al: This is a direct consequence of theorem 3.6(b) (i).
(b): Fix f € M . n
Then (pqf) is a Cauchy sequence with limit f. From corollary 2.19 it follows n q
I f (x) I = lim I [pqf] (x) I == lim I [Pqf] (E xl I $ n n q
q--- q-
liE xlln
s.....:.J.. n -1 q II pllfll <_ ~ lim sup ( q ) M q-+«> Irl! n
Because of the previous lemma M f n ~ 0, and M have reproducing kernels. n
o
The computation of these reproducing kernels requires the following results.
Lemma 3.10. Let f € M and let y € IRt2 . Then
Proof. Follows from property (1.6).
Lemma 3.11. Let y € nu2 • 2n
Then (K~(YIY»q is a convergent sequence in IR with limit II~I~
Proof. Let n ~ 1. From the proof of lemma 2.18 it follows
liE ylln (2n +q -2) (n -2 +q -1) ... (q -1) ---.,;q=..,.-_ (2n -2 +q) (2(n -2) +q) ••• (q) n!
_ II ,,11 2n So lim Kq(y,y) ~. The case n = 0 is trivial. n - n!
q-
o
o
- 39 -
Defini Hon 3. 12. The functions Kn' n :2: 0 and K from :m 1/,2 x :m 1/,2 into mare
defined by
n K ( )
_ (x.y) n x,y - n!
K(x,y) = exp(x.y)
Theorem 3.13. Let y Em1/,2 and let n :2: O.
Then we have
(a)
(b)
(c)
(d)
n Kn ( 0 , y) E M and II K (., y) II M = !Lx!L .
n n .frlT
For each f E M n
II ¥II 2 K ( • , y) E M and II K ( 0 , y) II M = exp (-2-) •
For each f E M
fey) = (f,K(o,y»M •
Proof (a): For q,p E IN we have
= Kq+P(y y) n '
From Lemma 3.11 it follows that (Kq(o y» is a Cauchy sequence. Because of n ' q
Lemma 3.9 lim Kq(o,y) exists pointwise and is an element of M • q~ n n
Let x E m22 , Since
we have
- 40 -
So with Lemma 3.11 and Lemma 1.11
lim K~(X'Y) = ~Ixllnllylln (II~II q-l'CO
y n 1fYII)
n 11 K ( )11 lim IIKq { )11 - lIyll n ., y M= nO, Y M---
q-l'CO Irl!
(b): Fix f EM. Then n
fey) = lim q~
lim q~
[pIf] (y) = n
(f , K~ ( 0 , y) ) M
(f,Kn(o,y»M
(c); From (a) it follows that
00
exp(O.y) EM.
2 So K(-,y) E M and IIK(·,y)II M= exp(II~1I ).
E Y q liE yll)
q
1 n = -, (x,y)
n.
(d) Since point evaluation in M is continuous and Af = K is the reproducing kernel of M.
$
n=O M it follows that
n o
Due to the fact that the elements of M can be approximated by harmonic func
tions in finitely many variables, certain properties of harmonic functions
carryover to the functions of M. The following two theorems are results of
this type. The first theorem is a generalization of the classical Mean-value
theorem for harmonic functions.
Theorem 3.14. Let f E M, let x E IR12 and let r > O. Then
f (x) lim 1
f f (.;) do 1 (.;> = . q~ C1 r q- q-
q-1 q-l S (E x,r) q
(3.14)
Proof. Put
a. q
1
- 41 -
f q-l S (E x,r)
q
f{~)do 1(~) • q-
The sequence (Pqf)q is a Cauchy sequence in M with limit f.
Since [Pqf] (x) = [pqf](E xl and since pqf is harmonic we have q
Ia. -[pqf](x) I = 1 I (f (~) - [Pqf](t;) )do 1 (~) I q q-l q-
o 1 r q- q-l S (E x,r) q
J
2 s 1 exp(tilL)lif - pqfll,.fOq_l u;;) q-l 2
o 1 r q- q-1 S (E x,r) q
So lim a. - [pqf] (x) o or q-+«> q
lim Ci lim [Pqf] (x) f (x) . q-+<x> q
q-+«>
The second theorem is a weak version of the min-max principle of harmonic
functions.
Theorem 3.15. Let D be a bounded open subset of IR~2 with boundary r such
that D is weakly closed in IRi2
. As for harmonic functions, we have
V D I f (x) I s XE
sup ncr
If (n) I .
Proof. Let r > 0 be such that D c {x ( IR~2 I II xii < r}. Then we have
2 V - If(x) I ~ eXP (2)lIfIl M XED
Since r = D\D, d = sup If(n) I < 00.
nEr Now suppose the assertion (3.14) were not true.
Then there exists y E D such that if(y) I > d. Let p > 0 be such that
:S;
:'
D
- 42 -
If(y) I = d + P. Since D is open there exists z E E D for sufficiently large q
q such that
(a) If(z) - fey) I < t p
(b)
From (b) it follows
1 < - p
3 for all XED .
We obtain
IcFlf](z) I = If(y) - fey) + fez) - fez) + CPqf](z) I ~
?: If(y) I - If(y) - fez) I - If(z) - [Pqf](z) I >
1 1 > d + p - 3 p - 3 p =
Further we have
sup ICpqf](n) 1'5 sup <If(n) 1+ ICpqf](n) - fen) 1< nEr nEr
1 <d+}p.
So we are in the following position:
There exists a function, viz pqf, on IRq which is harmonic and satisfies
ICPqf](z) I > d + t p and sup ICpqf](n) I < d + t p •
nEr
Since D is open in IRt2
, observe that EqD is open in IRq.
We have z E E D. q
We are ready if we can prove that the boundary of E D is contained in E r. q q
Now Eq is a compact operator and D is weakly compact in IRt2 , whence Eq(D)
is compact in IRq
Therefore E (0) is closed in IRq and E (D) c E (0). Since q q q
~\E (D) C {E (O}}\E (D) C E (D\D) q q q q q
the boundary r of E (D) is a subset of E (r). q q q
E (n q
(3.15)
- 43 -
This is contradictory to
I [Pqf](z) I > d + ~ p > sup nEE (f)
q
I [Pqf] (n) I 2: sup I [Pqf] (n) I . nEfq
o
Next we present two orthonormal bases in M. One of them arises from an expli
cit construction, the other is given explicitly. This construction is by in
duction and is based on a generating principle due to Muller. This principle
is formulated in the following lemma.
Lemma 3.16. Let f denote the restriction to sq-l of a harmonic homogeneous
polynomial p of degree m in q variables.
Let n 2: m. Let F: sq + ~ be defined by
F(/1 - t2~ + te 1) = Aq+l(t)f(~) q+ n,m
q-l ~ E S , t E [-l,lJ •
(For the definition of Aq+1 see (1.20).) n,m
Then F is equal to a restriction to sq of a unique harmonic homogeneous
polynomial of degree n in q+l variables.
Proof. See [Mu], definition 4 and lemma 15. o
Before we proceed we mention that the numbers dq n
dim (Hq) satisfy the rela
n tion
n
L m=O
as a consequence of corollary 2.7.
Definition 3.17. Let e O,l = 1.
Let {e 1,e 2} be an orthonormal basis for H2, n 2: 1. n, n, n Let n,i E IN with i > 2.
There exist unique q E IN such that
Because of the relation (3.15) there exist unique m,j E IN u {oJ with
o $ m < nand 1 $ j $ dq such that m
(3. 16)
- 44 -
n i = L
k=m+l
The function e 0: IRt2 + ~ is defined by n,~
E lx ~ a q+
e 0 (x) n,~
te 1 + 11 - t2~, t E [-1,1], ~ E sq-1 q+
a
So from {eO,l} U {en,i I n E IN,
{eO 1} U {e 0 n E IN, 1 ~ i ~ , n,~
E tX q+
1 ~ i ~ dq
} we obtain the set n
dq +1 } and so on. n
O.
{} H2. Theorem 3.18. For each n E IN let e 1,e 2 be an orthonormal basis for n , n, n, Let e 0' n,i E IN, i ~ 3 be constructed as in definition 3.17.
n,~
Then we have
(a) For all n,q E IN with
normal basis of ~. n
q ~ 2 the set {e 0
n,~ I 1 ~ i ~ d
q} is an ortho
n
(b) For all n E IN the set {e 0 i E IN} is an orthonormal basis of n,~
M . n
(c) The set {eo, 1 } U {e n, i
n,i E IN} is an orthonormal basis of M.
Proof. (a): The proof is by induction.
First we note that
mal basis of H2. for all n E IN the set {e 0 I 1 ~ i ~ d
2 } is an orthonor-n,~ n
n Next we assume that for all n E IN the set {e 0 I 1
n,~ is an orthonor-
mal basis of ~. n
Starting from this assumption we show that {e 0 I 1 ~ i ~ dq
+1
} is an ortho-n,~ n
normal basis of Hq+1 for all n E IN.
n Fix n E IN. Let il E IN with d~ < il ~ d~+l. As we have seen there exist unique m
1,jl such that e 0 is given by formula
n'~l
(3.16) in which we put i i1
, m = mt
and j = jl.
Because of lemma 3.16 e 0 E Hq+l. n'~l n
- 45 -
Because of lemma 1.12(b) for all x E IR q +1\{O}
IITt(n + ~2) L 1 ""2 n q+ x
( + 1 ) II xII A (-II -II .eq+1) r(n + _q ____ ) n,m x 2
So also the e . with 1 ~ i ~ dq are given by formula (3.16) in which we put nt~ n
m == nand j == i.
k . . l' i d q +1 Ta e ~1,12 E IN with ~ 1 1, 2 ~ n .
For j = 1,2 take the unique mn,jn E IN U {a} such that e . is given by .. .. . . n,l£,
formula (3.16) in which we put i == i£" m == m£, and J == Jt
.
We note that 1 ~ jt ~ dq
, t 1,2. mR.
Put a, (e .,e .) M" n'~l n'~2
Lemma 2.14 says
+ 1
f e i (ll)e . (n)dO" (n) n , 1 n'~2 q
Take variables t E [-l,lJ and ~ E sq-l such that n
and
dO" (n) '" (1 - t2
) 2 dO" 1 (~)dt q q-
2n
--,----,.--:"- r (n + 2n (q+1) /2
-1
1
f f q-l
S
Because of the induction assumption we get
Finally, Lemma 1.12 and Lemma 2.14 yield
te 1 + q+ . Then
- 46 -
a =
{I dq+1} ,~+1 Thus we have proved e . 1 ~ isis an orthonormal basis of n-n,~ n n
for arbitrary n.
(b): Fix n € IN.
Because u q~2
sis of M • n
tf. is dense in M , the set {e . i E IN} is an orthonormal ba-n n n,~
(c): Since M (i
n=O M this statement follows trivially from (b).
n
The second basis consists of homogeneous polynomials which are not harmonic
in general.
Definition 3.19. Let MI denote the set of multi indices
00
Let Is I = I j=O
Sj' s! = s 1 ! s2 ! •.. and a s,t IT <5
1 s.,t.
j= J J Put MI == {s E MI
n I lsi = n} for n ~ O. For each s € MI we define
q>: s IRR,2 ~ It by
lP (x) = s II
j=l
s. J (x.e . )
J
Put ~ = {q> I s € MI} and ~ = {q> I s € MI }. s n s n
Theorem 3.20. Let n E IN. Then we have
(a) Vs EMI : q>s c Mn· n
(b) The set ~ is an orthonormal basis of M • n n
(c) The set 4> is an orthonormal basis of M.
q Proof. (a): We first construct auxiliary harmonic functions gs,k with
s € MI and k,q E lN large enough. n Fix k E IN.
For 1 s j s k, q ~ k we define the operators D~,k E L(IRR,2) by
f m 0
q
o ::+1 1 s j - 1 D. ke . mk m ~ J, J+
j S m ~ q - 1
o
(3.17)
(3.18)
(3.19)
(3.20)
(3.21)
The
q Dj,keh
operators q
0, k J,
q 0, k maps
J,
Dq e = , k ' ), J
0,
have
Span
k 'i' q 2 L II D4 kxll =
j=l J,
E Dq+P = Dq q j,k j,k
- 47 -
h ,. j + Ink, 0 :5m:5q- 1 .
the following properties
{e, Ink I o :5 m :5 q - 1} isometrically onto IRq . J+
II xii 2 for all x E IRkq
.
k For all SEMI and k,q E IN with
n
g~,k: IRJl. 2 -+ (£ by
I SJ' -n and q ~ k we define the function j=l
k gq k(x) = s,
q q IT K (0, kx,e,)
1 Sj)' J
Because of (3.17) the j-th factor is a harmonic homogeneous polynomial of
degree s, in the variables x x x So gq is a harmonic ho-J j' j+k'···' j+(q-l)k· s,k
mogeneous polynomial of degree n in the variables x1
' ••• ,xqk
. k k
Claim: For S,t E MI , k E IN with I s, I t, = n, p ~ 1 and q ~ k we n j"'l ) j=1 J
have
Proof: Put (l
Then we have
(s!t!) ~ (l = --~--~--~
(21T) (q+p) k/2
Because of (3.20)
q q+p K
t (0, kx,e,) = j)' J
k q
II K (e"e,)6 t 1 tj J ) Sj' j
II xii 2
f IR (q+p) /k
2 k
q+p q+p q q II K (0, kx,e,)Kt (0, kx,e,)dx.
, 1 s, J, J ,), J J=) J
e
Application of the transformation of variables x -+ (Y1'.'.'Yk) with
y, = Dq+Px yields because of (3.19) J j,k
- 48 -
k - ( I
j=l
2 lIy.1I ) /2 J k
e
2 lIy.1I __ J_
q+p q IT K {y.,e.)K (y"e.)dY1, .. dYk
1 s, J J t. J J
j= J J
k = (s!t!) ~ IT
j=l [ 1 ~+p I e 2 q+p q
K (y.,e.)Kt
(y.,e.)dy.J s, J J . J J J
s!
IRq+p
k q
IT K (e.,e.)o t . 1 s. J J s" . J= J ] J
This proves the claim. k
Let <P € ¢> • Let k E IN with Is, = n. s n J j=l
We will prove that gq ~ <P , q ~ ~, in s,k s
From claim (3.21) it follows that for q
J J
M • n
~ k, P ~ a
== s! k IT
j=l
q+p K (e"e.) Sj J J
- s! k IT
j=l
q K (e.,e,)
Sj ) J
Because of Lemma 3.11 the sequence k IT
j=l (gq k is a Cauchy sequence in M • Let s, q n strongly to Dj,k € L(IR!2) defined by
{
e,
D, ke. mk = eJ
J, J+ m em+1
m = 0
1 ~ m < j
m ~ j
Kq
(e"e,» is convergent. Hence s. ) J q
J q 1 ~ j ~ k. The operator Dj,k tends
= a h ~ j + mk, m ~ a .
Now because of the proof of Theorem 3.13(a) we have
lim gq k (x) s,
q-+<»
k v'ST" II
j=l
- 49 -
s, J
(D, kX' e,) ), J
s , ! J
s, 1
k II
1 (x.e,) J = lP (x) •
J s
Hence qJ EM. s n
v'ST"
(b): Let qJs ,qJt E Mn' Let k f: IN such that k
I s. J
= lim s~ q-+<»
k II
j 1 K
q (e"e.)a
s. ] J s.,t, J J J
k IT
1 <5 = 0 s"t, s,t
J )
So ¢' is an orthonormal system in M . n n
k
L j=l
t, J
n. Then
Let f E M with f ~ Span(~ ). Let a E IRt2 and let q E IN. Since n n
we have
K (",E a) n q
1 n
n •• E a)
q Span(¢ )
n
feE a) = (f,K (o,E a»11 O. q n q IVI
So f(a) = lim feE a) o for each a E IRt2
• Conclusion f q-+<» q
Hence ~ is an orthonormal basis of M • n n
(c): Trivial.
o.
o
We give two other descriptions of the inner product <.,.> restricted to sub
spaces of M. The space Span(~) consists of all polynomials in a finite but arbitrary num
ber of variables.
- 50 -
Theorem 3.21. Let f,g E Span(~). Then
(f,g)M = [f(a)g](O) •
Proof. For all s,t E HI we have
[tps(a)tPt](O) = 0 s,t and
o
Since ~ c Span(~) this theorem is a generalization of Lemma 2.25.
Next we describe the inner product
sentation R -+ ~ (R) I If!' REO (IR q)
n
( ., .) M restricted to H. Because the repre
is irreducible and because If! is finite n
dimensional, we have
( { 1 { '} n I q}). {:l Span -- •• Re1
+ l.Re2 R E O(IR } tf-rn: n
(The function _1_(o.e1
+ ie2
)n is called a cyclic vector.) Hence rn!
1 . )n o (IR 9,2) }) = H Span ( {--( •• Re 1 + l.Re2
R E
In! n and
1 + iRe2
) n I R E O(IRg, )}) H . Span ({-( •• Re1
n 2: 0, = rn! c
Lemma 3.22. Let n,m E IN and let a,b,c,d E IRR. with a .l b, c .l d and c
II all = II bl! = II ell = II dll. Then
(_1_( •• a + ib) n, _1_( •• c + id) m) M rn! v"nf
(c + id. a + ib) n IS n,m
~. We restrict ourselves to the case n = m. Put
Because of Theorem 3.21 we have
= { L j=l
(c. + id.) (a. - ib.»n J J J J
(c + id. a + ib) n . o
- 51 -
Theorem 3.23. Let f,g E H be such that
f and g
n. (f,g)M= a j St (Ste 1 + iS t e 2 • Rj e 1 + iRj e 2) JOnj,m
t
Proof. Apply Lemma 3.22. [l
Next we present a unitary representation IT of the full orthogonal group
O(IRt2) in M which is a natural extension of the representation ITq
of O(IRq
)
in ~.
Definition 3.24. The mapping IT: R ~ IT(R) is defined by
* TI(R)f = foR,
Theorem 3.25. The mapping TI is an unitary representation of O(IRt2 ) in M.
Proof. Remark that R* -1 R for R E O(IRt2). We prove conditions (a) and (b)
of theorem 1.4.
(a): Fix x,y E IRt2 and R E O(IRt2). Then
K(Rx,Ry) = exp(Rx.Ry) exp(x.y) = K(x,y)
(b): Fix y E IRt2 , Y f 0, and € > O. We recall that lexp(x) - 11 ~ 21xl for
all x E IR with Ixl < ~. For R E O(IRt2) with
IIR - III < min{1/2I1ylI2,e:/llyIl2exp(2I1YI!2)}
we have
2 2 IIK(-tRy) - K(·,y)II M = 2[exp(lIyll ) - exp(Ry.y)]
= 2 exp(lIyIl2)[1 - exp(Ry - y.y)] ~
~ 2 exp ( 211 yll 2) II yll .11 y - Ryll < € •
From Theorem 1.4 it follows that IT is a unitary representation. D
(3.22)
(3.23)
- 52 -
Lemma 3.26. Let n ~ 0, let R € O(IRt2). Then we have
IT(R)P =PIT(R) n n
Proof. Let W =: Span({K(o,y) I y ( IRt2}). For y € IRt2
n IT(R)P K(.,y) =: IT(R) (o.y)
n n!
n (o.Ry) =: P IT(R)K(o,y)
n: n
Hence IT(R)P = P IT(R) on Wand hence on the whole of M. n n
So M is an invariant subspace of the representation TI. n
Theorem 3.27. Let n,m ( IN, n # m.
(a) : The representation R-+ IT (R) 1M' R € O(IRt ), c n
(b) : The representation R -+ IT (R) 1M' R ( O(IRt2),
n
is
is
(c) : The representations R -+ TI (R) I M and R -+ 11 (R) 1M' , '1 '1 n m not un~tar~ y equ1va ent.
irreducible.
irreducible.
R (O(IRt), c
~. (a): Let V, V # {a} be a closed subspace of Mn' such that
VREO (IR1) IT(R) V c V, i.e. V reduces each TI(R). c
o
are
Let P be the orthogonal projection from Manto V. Since Vi n
M €I V is also n
an invariant subspace it follows that for all R ( O(IRi2)
TI(R)P = PTI(R) •
Fix q ~ 2.
Since O(IRq
) c O(IRt2), the preceding statement yields
TI (R) P =: PIT (R) for all R E O(IR q) •
The mapping R -+ TI(R) I~' R E O(IRq
), is an irreducible unitary representation,
n see section 2, so for all R ( O(IRq ) •
The operator pqppq maps ~ into Hq and from (3.22) and (3.23) it follows that n n n n
for all R E O(IRq)
- 53 -
Because of Schur's Lemma, see Lemma 1.6, there exists A ;::: 0 with
pqppq = A pq. q
n n q n
From pq+1 ppq+1 = n n
pq it follows that n
A A q q+l
So we have proved that there exists A ;::: 0 with
q ~ 2
Now pq tends strongly to P and so n n
p = )'P n
Since V ~ {a} we have P = P . n
This means V = Ai • n
(b): This is a corollary of (a) because O(IR~c) c O(IR~2).
as well assume n < m. (c): We may
Suppse fI I M n
and filM were unitarily equivalent representations m
Then there would exist a unitary operator U: M + M such that n m
Fix q > 2.
We define the representation rrq : O(IRq
) + L(M ) by n n
rr~(R) = TI (R) 1M ' n
Similarly the representation rrq
is defined. m
For fixed k € INO IN U {a} and SEMI with 2k + lsi ~ n the subspace Vk,s
of M is defined by n
Vk
= {f E M I ,s n ~ n-2k-lsl
h (E x) U E xlI2k q q
By rather straightforward arguments it can be shown that
( 1) k ~ k' or s ~ s'
"" s. rr (x.e +.) J}
1 q J
(2) The subspace Vk
is rrq(R)-invariant for all R E O(IRq
) and the represen-,s n
tation rr~lv is irreducible. k,s
(3) M n
- 54 -
(4) dim(Vk,S} = d~-2k-lsl •
Since U is a unitary operator which sets into equivalence representations
rrq and rrq we have both n m
M m
and
$ UVk (k,s) EINOxMI,2k+ls Isn ,s
pqUV # {a} for some k,s . m k,s
Moreover the space pqUVk must be invariant under IT m ,s m
This is contradictory to dq 2k I I # dq
• n- - s m
whence pqUV m k,s = ffI.
m
The representation of definition 3.24 resembles the representation introqu
ced in [Ba 3], section 3.
* One may think that M , n ?- 2 only contains functions of the form 9 0 R for n
some 9 € ~ and R E O(IRt2).
The following example shows that this is not the case.
Define f: IRt2 + II: by
<X>
f (x) \' 1 n L -;-(x. ej
) j=l J
00
o
Then f € Mn and II fll M ( \' 1) ~ 1"1 h' ,.a d ( ) L :2 Yn:. Assume t ere ex~st 9 € n~ an R € 0 IRt2
with * f = 9 0 R
=: Re .• Then J
j=l j
I f (x) I n
«x.a l ) , ... , (x.a» 2 2 2 = g(--......;;;"..2-----..
q-2--=-1:! «x.a l ) + •.. + (x.aq ) ) .
«x.a l ) + .•• + (x.aq »
So there exists B > 0 such that
2
If(x) In ~ B«x.al}2 + .•• + (x.a
q)2) •
Then
2 2
(T> n I f ( e j) In:,; S « e j . a 1 ) 2 + ... + ( e j • a q) 2)
(3.24)
(3.25)
(3.26)
(3.27)
- 55 -
Since
co
jt (3«ej.a
1)2 + ••. + (e j .a
q)2) = (301a
1U
2 + •.. + lIaq l12
) Bq
2 co
and since L (~)n is divergent for n ~ 2 we arrive at a contradiction. It j=l J
is not hard to see that indeed
Finally we introduce orthogonal projections T(p) on M where P is an orthogo
nal projection in IR~2'
Fix an orthogonal projection P on IR~2'
Let W denote Span ({exp (-. y) lye IR ~2}) •
The operator T(P): W + W is defined by
m m T(P) ( I a.exp(o.y.»
J ] I j=l
a.exp(-.Py.) , ] ]
For f,g E W we have
T(P)f = f ., P
(T(P)f,g)M = (f,T(P)g)M
II T (P) fll M :;:;: II fll M •
m € :IN, a j E It, Y j E IR Q,2 .
From (3.27) it follows that T(p) has a unique continuous extension to M. The unique continuous extension of T(p) to M, again denoted by T(p),satisfies
the relations (3.24) to (3.27) with f,g € M. Because of (3.26) and (3.27),
T(p) is an orthogonal projection. The spaces T(p)M and T(PlM have the repro-1 n n
ducing kernels n:(Px,y) and exp(Px~y), (x,y) E IRQ,2 x IRQ,2' respectively.
- 56 -
4. A space of functions on the unit ball in IRt2
In this section we introduce a space of functions on the unit ball in IR~2'
The following notations will be used.
B the unit ball in IRt2 •
B (a,r), the ball with centre a and radius r in IR R,2 •
We define another inner product in M.
Definition 4.1. The inner product (.,.) on M is defined by *
00
(f,g)* = L ~(P f,P g)M' f,g EM. n=O n. n n
The corresponding norm will be denoted by 11.11*,
Lemma 4.2. Let x E B.
Then the following results are valid
(a) For all f E M n
I f (x) I s II xII nil fll *
(b) For all gEM
Proof. (a): Fix f € M . Because of Lemma 3.6 we have n
n I f (x) I s ~I filM == II xii nil fll * .
rnT
(b): Fix g € M. From (a) it follows that
co 00
I g (x) I S; L I [P g (x) I S; I n=O n
n=O
S ( 1 2) ~I gil * 1 - II xii
II xII ~I Pngll* S
Lemma 4.3. Let (fk)k be a Cauchy sequence in M in II .II*-sense.
o
(a) Then for each x E B the sequence {fk(x»k is a convergent sequence.
(b) Then the function f: B ~ ~ defined by
f(xl = lim fk1x) , k-+oo
- 57 -
X E B
satisfies the following conditions:
(i) For each x E B
(ii) The function f is continuous on B.
Proof. The proof is similar to the proof of theorem 3.6.
Definition 4.4. Let N denote the Hilbert completion of M with respect to
II .11* consisting of all functions on B which are pointwise limits of
Cauchy sequences in M in 11.11 -sense. *
[1
The corresponding inner product and norm of N will be denoted by ("')N
and II .11 N'
We make some remarks about N.
(4.1) The elements of M will be identified with their restrictions to the unit
ball B.
So ffI eN, MeN etc. n
(4.2) Since II.II M and II.II N are equivalent on Mn the closure of Mn in N is Mn itself.
(4.3) Lemma 4.2 yields the following inequality:
For all fEN and x E B
I f (x) I ~ 1 ) ~II fll 1 _ II xii 2 N
(4.4) Because of (1.8) the space N has a reproducing kernel.
Definition 4.5. The functions Lq
, Land L: B x B + ~ are defined by n n
L (x,y) n
L(X,y)
n (x. y)
1 1 - (x.y)
x,y E B
- 58 -
'!'he next theorem states that the functions Lq , Land L are the reproducing n n
kernels in (.,.)~sense of the spaces ffl, M and N respeeti vely. n n
Theorem 4.6. Let x E B.
(a) Then L~(O,X) E ~, IIL~(o,x)IIN = (n!K~(X'X»~ and for all f E ~
f (x) = (f, L ~ (. , x) ) N
= II xII n and for all f E M n
(e) Then L(o,x) EN, IIL(.,x)II N ( 1 )~ and for all fEN 1 _ IIxll2
f(x) = (f,L(o,x»N •
Proof. (a): Fix f E ~. Then n
f(x) = (f,K~(.'X»M = (f,L~("x»N •
Further we have
(b): Similar to (a).
(c): From (1.8) and (b) it follows that the function
(x,y) -+ I n=O
n (x.y)
is the reproducing kernel of N.
1 1 - (x.y) ,
The remaining part runs similarly to (a).
(x,y) E B x B
Next we present classes of functions which are in N. Recall the definitions
of the space Harm(~), of the projection Dq and of the mapping Hq
which can n
o
respectively found at the beginning of section 2, after corollary 2.4 and in
definition 2.8. We identify f E Harm (Bq ) with x -+ feE x), x E B. q
Theorem 4.7. Let q ~ 2.
The space Harm(Bq (O,I:2» is contained in N.
(4.5)
- 59 -
Proof. Let f E Harm (Bq (0,12». Put g =
f = ~g on Bq(O,r). So
f I q-l' There exists r > 12 such that S
00
II Dqgll 2 p 2n < 00 for all p wi th 0 ::; n q-l
S p < r L
n=O
and
00
f (x) L n=Q
We have
00
IIfllN = L II If1 (D~g) II ~ n=O
00
I .!., II If1 (Dqg) II ~ n=O
n. n
2nr(n + Si)
II Dq
gll2
1 co 2
= I 2 q/2 • n=O
7f n. n sq-
because of lemma 2.14.
Choose p between and r. n q
2 r (n + "2) (_1) 2n Since is uniformly bounded in nand
n~ p we find II £II N < 00.
00
II Dqgll 2 2n n q-1 P
S L
n=O
Example. Fix a > 1. Consider the function f : B ~ ~, defined by a
f (x) a
co
I Im«x.e1
) + i(x.e2»m
m=O a X E B .
< 00,
The series in (4.5) is pointwise convergent on {x E IR 12 I II xii < a}. The
Since
function f I is harmonic. a B2(O,a)
II 1 «) ID.I 1 ( . ) m 2m/ 2m m .. e 1 + i ( •. e 2» I N = 2m e 1 - ie 2 • e 1 - ~e 2 = a , a a
it is easy to see that fiN for 1 < a ~ 12 and that fEN for a > 12. a a
o
Since the harmonic functions of M are dense in M in II .11 N-sense, the elements
of N can be approximated by harmonic functions in finitely many variables.
Therefore we get the analogues of theorems 3.13 and 3.14.
- 60 -
Theore~ 4.8. Let fEN, let x E B and let r > 0 be such that B(x,r) C B.
Then
f(x) = lim __ 1 __
q-1 q-l><» 0 1 r q-
J q-1
S (E x,r) q
f ( ~) do 1 (~) • q-
Proof. Analogously to the proof of theorem 3.13.
Theorem 4.9. Let D be an open subset of B with boundary r such that D is
weakly closed in Band DeB. As for harmonic functions we have
v I f (x) I XED ::;; sup If (n) I
nEf
Proof. Similar to the proof of theorem 3.14.
From theorems 3.16 and 3.18 we obtain two orthonormal bases in N namely
n,i E IN}
and
{rnT" (j) I n E IN U { 0 }, SEMI } • s n
Finally we present a unitary representation of O(IRt2' in N.
Definition 4.10. The mapping E: O(IR~2' + L(N) is defined by
E (R) f
Theorem 4.11.
* = foR
(a) The mapping E is a unitary representation of O(IRt2
) in N. (b) The space Mn is invariant under the operators E(R), R E O(IRt2).
(c) The representation R + E(R) 1M ' R E O(IRtc )' is irreducible. n
Proof. (a): Apply theorem 1.4.
(b) and (e): See lemma 3.26 and theorem 3.27.
o
D
n
- 61 -
5. Spacial isomorphisms between M, the symmetric Fock space, the Bargmann space IN
and L2 (IR ,dl1)
In the next three subsections we deal with the subsequent cases as mentioned
in the above title.
5.1. The symmetric Fock space
(5.1)
Let X denote a separable Hilbert space over the field IK (with IK = IR or
IK ~) and let (u,>, J denote an orthonormal basis in X where J = {l, ••• ,m} J )E
or J :::: IN.
Fix n ~ 2.
Let Tn(X) denote the space of continuous multilinear functions mon Xn {i.e.
m( xl' ••• , (lX, + y" ... , x ) :::: 0 m{ xl' ... , x ., ..• , x ) + m( x I ' ... , y , , ... , x ), J J n J n J n
1 ~ j s n) with
In Tn(X) we define the inner product (.,0) by Tn(X)
:::: L n
m(u, , •.. ,u, )t(u, , ... ,u, ), m,t E T (X) • J 1 I n J 1 I n j1,···,jnEJ
The space Tn(X) is a Hilbert space and is called the n-fold Hilbert tensor
product of X. n
For Yj
€ X, 1 ~ j S n, we define ® j=l
n n
n Y,: X -+ ([ by
J
[® y.](x1,···,x) j=l J n
II (xj'Yj)X' j=l
n The multilinear n-function ®
n Yj is an element of T (X)
j=1
n n n ( ® Yj , ® Z ,l = II (z.,y,) X . j=l j=l ] Tn (X) j=l J ]
and
n (5.2) The previous result yields that the set { ® u
m, I m € I
n } is an orthonormal
basis of Tn (X) . j=l J
- 62 -
Let S denote the group of permutations of {1,2, .•. ,n}. For each cr E S we n n
define the unitary operator 6: xn + xn by
(x x) I cr{l) , ••• , (J(n)
The mapping m + m 0 a, mE Tn(x), is a unitary operator from Tn(X) into
Tn(X) .
Hence the operator S : Tn(X) + Tn(X) defined by n
is an orthogonal projection in Tn(X) and is called the symmetrizer of Tn(X) •
We define the Hilbert spaces Sm(X), m ~ 0, by
sO (X) II(
S1 (X) X
Sn(X) = S Tn (x) n '2: 2 • n
Observe that Sn eX) equals {m E Tn(X) I VaES : m = m 0 3} for n ~ 2. n
The Hilbert space Sn(X), n ~ 0, is called the n-fold symmetric Hilbert ten-
sor product of X.
The symmetric Fock space S (x) is defined by
OC>
S (X) = '9 Sn (X) . n=O
The notation S (X) has been taken from [Gui].
The inner product and norm in SeX) will be denoted by (.,.) S (X) and II .11 S (X) .
Elements of SeX) will be denoted by the capitals F, G, etc. or (F) with n n n
F E S (X). n
The symmetric Fock space SeX) is a real (complex) vector space if X is a
real (complex) vector space. If X is a real vector space we denote the com
plexification of Sn(X) and SeX) by §flex) and Sex) respectively.
Next we present an orthonormal basis in Sex). Therefore we recall definition
3.10.
IN u {OJ.
- 63 -
Definition 5.1. Fix n E INa.
Put IN 0 J {s E: MI I v, E IN \J: s, ; J J
a}. For each SEMI n INa we n ;J
define ~ E Sn{X) by s
~ s
{n;)~S ( 0 s. n,
JEJ u, 0 ••• 0 u,)
J J s,-times
J
Lemma 5.2. Fix n E: INa.
Then we have
(a) The set {~ s
(b) The set {~ s
sEMI n INa } is an orthonormal basis in Sn{X) . n ;J
SEMI n INa } is an orthonormal basis in S{X). ;J
Proof. (a): Let s,t E MI n INa . We can write n ;J
~ s
'L n {n,')'S ( 0 )
u'" (J') s. n, 1 ... J=
and I ~ n
~ = (n.,) S ( 0 ) t t. n, 1 uS{j) ,
J=
where a and S denote suitable non decreasing functions from {1, ... ,n} in IN
with a (j) £ iff
£-1
~ sk < £ $
k=l
Then
£
L sk etc. . k=O
n n n! (0 S 0 )
~ ua{J')' n uS{j) S{X) (s!t!) j=1 j=1
1 L
(s!t!)~ OES n
n
IT (Ua{j) 'US{O{j»lX . j=1
Now (~s'~t)S{X) f a iff s = t.
We see that
Hence {~ I SEMI n INa } is an orthonormal system in Sn{X) . s n ;J
Because of statement (S.2) this system is an orthonormal basis of Sn{X). 00
(b): Because S{X) = e Sn(X), (b) follows from (a). n=O
[J
- 64 -
Definition 5.3. For each x E X we define EXP (x) E SeX) by
@n @n
n EXP (x) (~) where x 0 x .
v'n! n j=l n.
2 2 We have II EXP (x) II S (X) = exp <II xII X) .
Let F E S(X). Then the function UXF: X + ~ is defined by
(5.3) For each sEMI with I j€J
s, J
x eX.
lsi we have
1 s, [Ux'¥sJ (x) = --L II (X,U
J,)/,
(s!)""2 jEJ
Theorem 5.4. Define UIRt 2
X EX.
(a) The operator '" is a unitary operator from S(IR£2) onto M.
(b) The operator maps SO (IR £2) onto Mn'
Proof. We replace (u,). J by the orthonormal basis (e')j in IRt2 . The set ----- ] J€ J {'¥s I SEMI} is also an orthonormal basis of S(IR~2) .
Fix sEMI.
Because of statement (5.3) we have for all x E IR£2
~ 1 co s. [UIR£ '¥ ] (x) = --~ II (x.e.) J =
s s (s!) j=l ] tp (x)
s
The set {~s I SEMI} is an orthonormal basis in M. Now the statements (a) and (b) follow easily.
Corollary 5.5.
Span ({EXP{y) lYE IR£2}) is dense in S(IRt2
) .
o
- 65 -
Proof. We remark that [UIR~ EXP(y)J(x) = exp(x.y) for all x,y E IRt2
, 2
Because Span({exp(o.y) lYE IRi2 }) is dense in M,
Span ( {EXP (y) lyE. IR ~2 }) is dense in S (IR £2)
This implies
Another proof can be found in [GuiJ, Chapter 2, proposition 2.2.
5.2. The Bargmann space B
The Hilbert space ~~2 is the space of square summable sequences in ~ with
inner product
(z,w)
o
The space ~£c is the subspace of ~£2 consisting of all sequences with only a
finite number of non zero entries.
Similarly to the embedding of IRq into IRt2 of section 3 we embed ~q into ££2'
We also embed IR£2 into ~£2 the latter space is then considered as the com
plexitication of the former. (The same notations for the inner product and
norm will be used.) We get the diagram
(5.4) The above mentioned embeddings give rise to
(5.5) Let Eq denote the orthogonal projection from ~t2 onto ~q.
Let Bq denote the Hilbert space U S(~q) with inner product a:q
(f, g)
Bq
and let B denote the Hilbert space U~£2S(~t2) with inner product
- 66 -
-1 -1 (f,g)B = (Uq;R, f,UCCR, g)S(q;R, ) ,
222 f,g E B •
The spaces Bq and B are called the Bargmann space of order q and the Bargmann
space of infinite order, respectively. With the aid of relation
embed Bq into B. Each f E gq will be identified with UCJ:~ [U-1fJ
2 a:q calently with z ~ f(~z), z € CC~2'
Definition 5.6. Let sEMI.
The function u : ctt ~ CJ: is defined by s 2
u (z) s
1 DO s,
--IT ( )J ~ z.e, (s!) j=l J
(5.4) we can
E B or equi-
We mention the following results which also can be found in [Ba 1J and (Ba 2J.
(5.6) The set {u I I SEMI n lNO
'{1 }} is an orthonormal basis of gq and S a:q , , ••• ,q
the set {u I SEMI} is an orthonormal basis of B. s
(5.7) The spaces Bq and B have reproducing kernels (z,w) ~ exp(z.w), z,w E q;q and
(z,w) ~ exp(z.w) I z,w E Ct2, respectively. The latter result will be proved
in corollary 5.11.
(5.8) The e~ements of Bq are analytic functions on a:q and the inner product in Bq satisfies
(f ,g) Bq f
2 exp(-lIzll) f(z)g(z)dz
1Tq
(5.9) The space U qElN
Bq is dense in B.
f,g E gq .
(5.10) The elements of B are analytic functions on CJ:~2' This will be proved in Lem
ma 6.1. The inner product in B satisfies
f exp (-II z1l2) f (z) g (z) dz 'lf
q f,g E B .
Observe that, unlike in M, the inner product of two functions in B is a limit
of ordinary integrals.
- 67 -
(5.11) The orthogonal projection T from B onto gq is described by q
T f q
fOE q
fEB .
Fix fEB. Then (T f) is a Cauchy sequence in B because of the relations q q
and
II T f - Tqfll 2B q+p
li:q
Before we define a unitary isomorphism from M onto B we need some auxiliary
results concerning special elements of M.
Lemma 5.7. Let a,b E 1i:£2' Then
(a) (o.a)n E M
(b) n
( ( 0 • a)
n: n
( 0 .b) ) n: M
(b.a)n n:
Proof. Since dim(Span({Re a, lm a, Re b, lm b})) ~ 4 there exists an orthogo-4
nal operator R E O(IR£2) such that a ' = Ra, b ' = Rb E ~ .
(a): Since each homogeneous polynomial of a finite number of variables is an
element of M, we have (o.a,)n E M, hence (o.a)n = (RTo,a,)n E M.
(b): Because the inner product is invariant under orthogonal transformations
we observe that
1 n 1 n (-, (o.a) , -, (o.b) )M n. n.
Corollary 5.8. Let a E ~£2. Then
exp (0. a) ( M and
1 n (-,) (b.a) n.
II exp( 0 .a)1I M
Theorem 3.21
because (b. a)
2 (II all ) exp -2-
(b I • a I ) • o
- 68 -
The elements exp(o.a) are called coherent states.
Each function f: IRt2 ~ C which belongs to M can be extended to a function
from ~~2 into ~ with the aid of the functions exp(o.a), a € ~t2'
The unitary isomorphism between M and B which will be defined below shows
that each function f € M can be extended to a function f from ~t2 into ~ be
longing to B. Thus M and B can be identified. This identification induces an
identification between S(IRt2
) and S(IRt2) = S(~t2)' See also [Gui], remark
2.20.
C~2 Definition 5.9. The function U
1: M ~ ~ is defined by
Z E <r:t2
, f Eo M •
A polynomial p from IR~2 into ~ can be extended to a function from ~£2 into
C replacing x by z. The next theorem yields that the two mentioned extensions
are equal.
Theorem 5.10. The following statements are valid:
(a) For each f E M and x E IRt2
(b) For each polynomial h and each z E ~t2
h(z) = [U1h] (z)
(e) The operator UI
is a unitary isomorphism from M onto B.
Proof. (a): For f € M and x € IRR,2 we have f(x) = (f,exp(o.x»W Thus the
result follows.
(b) and (c): Let SEMI. Let n,q € IN with lsi = nand s = a for all p > q. p
Let z € C~2 and let w = I zJ.ej
• The natural extension of ~s to a function j=l
from ~12 into ~ is us'
Since ~ = ~ (E .) and since ~ E M we have s s q s n
u (w) s
theorem 3.21
u (z) • s
- 69 -
The operator U maps the orthonormal basis {ttl I 5 E MI} of M onto the ortho-1 s normal basis {u I SEMI} of B. SO (c) follows immediately.
S
Each polynomial h is a finite linear combination of I.P , SEMI. s
So for all z E ~t2
o
Corollary 5.11. The function (z,w) + exp(z.w), z,w E ~t2 is the reproducing
kernel of B.
Proof. For all x,y E IRt2 we have
exp (x.y) I SEMI
Fix Y E IRt2 and w E ~t2'
Then
I.P (y) I.P (x) . S S
exp(w.y) [Ulexp(o.y)](w)
= I I.Ps(y) [ U1
I.Ps ](w) SEMI
I SEMI
Complex conjugation yields
exp(y.w) = I SEMI
I.P (y) u (w) . S s
u S
Fix z E ~t2' Analogously we get
exp(z.w) = [U1exp(o.w)](z) = I
SEMI u (w)u (z)
s S
From property (1.9) it follows that (z,w) + exp(z.w), z,w E ~t2' is the repro-
ducing kernel.
See also [Ba 2J, p. 201.
The inverse U~l links to each fEB the restriction of f to IRt2 . Next we
present an isometry from B into M. Let Te and TO denote the orthogonal projections on IRt2 which satisfy
o
(5.12)
- 70 -
nEIN • o
Observe that Te + TO I.
IR £2 Definition 5.12. The function U
2: B -~ «: is defined by
T + iT f« O e» x , X E IR i
2, fEB .
Theorem 5.13. Let fEB. Then we have
(a) The function [U2fJI 2 is harmonic. IRq
(b) ([U2£] I 2) IRq
is a Cauchy sequence in M with limit U2f.
(c) The operator U2
is an isometry from B into M.
Proof. (a): For all (xl"'" X2q
) C IR 2q we have
, ... , X
2q_
1 + iX
2q ,0,0, .•• ) .
Ii
Since f is analytic on ~q, we observe that U2
f l 2 is harmonic. IR q
(b): For each q E IN we identify the functions [U2f]1 2 and [U2f](E 2 0). IR q q
See the remark proceeding definition 3.2.
For each q E IN we have
1I[U2f](E2q')II~ = J
2 exp(- ~)
1 [U2f] (xl' ... ,x
2q) 12 2 dx
(2'lT) q IR
2q
+ iX2
+ iX2q
II xii 2
J x x
,O, ••• } 12 exp(- --)
1 f ( 1 2q-l 2 = I ••• ,
IR2q
Ii 12 (2'lT) q dx
J 2 2
1 ;v ,0, •.. ) 12 exp ( -II ull -II vii ) dudv f(u 1 + ivl,···,uq + ~
q 'lTq
- 71 -
2 exp (-II zll 2) ,0, .•. ) I dz
nq
For q,p E IN we have
Because of (5.11) the sequence (f(E 0» is fundamental in B. Hence q q
([U2fJI 2) is a fundamental sequence in M. The limit is U2f. So (b) is IR q q
proved.
(c): From (b) it follows that for each fEB
So for each fEB
IN 5.3. The Hilbert space L
2(IR ,d~)
II fll ~ because of 5.10 and 5.11 0
a Gaussian measure
In this subsection we introduce the Hilbert space (IR IN ,d~) based on the
papers of [Ko] and [OrJ. We supply the space of finite sequences IR~ with c
o
the inductive limit topology induced by the Hilbert spaces IRn. Then IR£ is c
a countable inductive limit of nuclear spaces. Hence IR~ is nuclear. See c
[Tr], Proposition 50.1. The space of continuous linear functionals on IR£ c
will be denoted by IR£f. The dual IR~' can be represented by the nuclear c c
'h IN Free et space IR
Indeed for each l E IR£' there exists x E IRIN such that for all ~ E IR~ e c
(t:. x)
and vice versa. So we have the triple
Define X: m i + m by c
2 x(~) =exp(-IIt;;1I /2) ,
•
- 72 -
t;. E mi. c
The function X satisfies the following conditions:
- X is positive definite
- X is continuous with respect to 11.11
x(O) = 1.
Following the Bochner-Minlos theorem, See [HiJ, Theorem 3.2 there exists a
probability measure ~, called a Gaussian measure on (mIN,B(m IN» such that
for all ~ E m t c
exp[i(l;.x)]dl!(x) •
Here B(mIN
) is the a-algebra generated by the family of all cylindrical Borel
sets B in mIN, i.e. B = {x E mIN ! «x.d1
) , ••• ,(x.dn
) E D} for some ortho
normal set (dj
);=1 in mic and a Borel set D in IRn.
For each R E O(mi ) the Gaussian measure is R'-invariant, i.e. l!{R'B) = ~(B) INc
for all B E B(m ).
IN We consider the Hilbert space L
2(m ,dl!), also denoted by £2'
Lemma 5.14. Let (d1
, ••. ,dq
) be an orthonormal system in lRtc and let
f E L1 (lRq
,exp(-lIuIl 2/2)du) .
Then the function F defined on mIN by
F(x) = f«d1.x), .•• ,(dq.x» , IN
x c lR
satisfies
F(x)dlJ(x} =
Proof. See [OrJ, § 1
I 2
f(u)exp(-lIull /2) du • (21T) q/2
o
(5.13)
(5.14)
(5.15 )
(5. 16)
- 73 -
For the introduction of an orthonormal basis in £2 we employ the Hermite po
lynomials.
Let H denote the Hermite polynomial of degree n. n
Then the H 's satisfy the following relations (cf.: [MOS], Chapter V, Secn
tion 6) :
2,
2 d -T -d [e H 1 (T) ]
T n-
00
2 -T
-e H (t) n
;r e-T2H (tlH (T)dt
m n _00
Definition 5.15. Let SEMI.
n > 0
T ( lR
n > 1, T E lR
n ,m 2: 0 •
IN The function ~ : lR ~ ~ is defined by s
00 • x)
lj! (x) 1
IT ) h Is Is:
H s s. 12 j=l J
Note that e. E IRt for all j E IN. J c
Theorem 5.16. The set {~s I s ( MI} constitutes an orthonormal basis of £2'
Proof. See [Hi], theorem 4.2. o
In definition 3.19 we have introduced an orthonormal basis ~ in M. The follo-
wing theorem presents the relation between ~ and the orthonormal basis
~ = {lj!s I s ( MI} of £2'
Theorem 5.17. Let sEMI and ~ ( lRt • c
Then we have
tPs (1;) 2 exp (-II 1;11 /2) f exp (i;. x) ~ (x) dp (x) •
s
- 74 -
Proof. Since exp(E;;.*) E £2' the integral in (5.16) is well defined.
Take q E IN such that t; E IRq and s 0 for all p 2: O. Put q+p
(l = f exp(t;.x) Ws (x}d\.l(x) •
IRIN
From Lemma 5.14 it follows that
J q 1 (e .• x) -II xII 2/2
(l = exp (t;. x) III -:;:;;;;:=== H ( J ) e dx IS. sJ' 12 (2'!T) q/2
q 12 J s .: IR J
q = II
j=l
q II
j=1
1 f exp(t;.x. -I2iT ) J
IR
With the aid of relation (5.14) partial integration of the factors yields
(l = ~ [~ J exp(TI2t;. -j=l 'IT J
IR
Since HO (T) t;~/2
e J we get
s. t;~/2 /11;11 2/2 q J q s. 2
II e J e II (e .• t;) J ellt;11 /2cp (t;) (l = :;
j=l IS:! 1ST j=1 J s J
IRR. Definition 5.18. The operator U
3: £2 + ~ c is defined by
e -II 1;11 2/2 J e(l;·x)F(X)dl.l(x) , t; E IRtc' F € £2 •
IRIN
o
An element f of M is completely determined by flIRt' So the following theoc rem is not so surprising.
(5.17)
(5.18)
(5.19)
(5.20)
- 75 -
Theorem 5.19~ The operator V3 is a unitary isomorphism from £2 onto M.
Proof. Because of theorem 5.21 the operator V3 maps the orthonormal basis ~
of £2 bijectively onto the orthonormal basis ~ of M. 0
In [KO] two irreducible unitary representations of the infinite dimensional
motion groupG~ of IRIN in £2 are introduced. The group Goo is the set
o (IR t ) x IR t wi th group operation c c
(R,l';) (S,n) =: (RS,Rn +~) •
In [Ko] the unitary representations T,S: Goo + L(£2) are defined by
[S(R,t;;)F](x) 2
e-(IIt;;11 +2(.;.x»/4F (R- 1 (X + 1;», x E IRIN FE £2'
(R,!;) E G 00
For each (R,';) E G put 00
and
~ ,.."
From the foregoing it follows that 7' and S are irreducible unitary represen-
tations from G in M. 00
For all R E O(IR~ ) we have c
,.." ,...., T(R,O) S(R,O) =: IT(R)
where IT is the unitary representation of O(IRt2
) in M introduced in section 3.
We will extend T and S to O(IRt2
) x IRt2
.
The unitary operator V1
, introduced in definition 5.9, will be used.
Theorem 5.20. Let (R,!;) E G and let f E M. 00
Then we have for all ~ E IRt • c
[T(R, .;) ~-( ~) exp(-2
[S(R,t;;)f] (I:;) exp(- Jl:Q -2
-1 i(1:;,~»[V1f](R I:;
2 ~} f (R- 1
(I;; + 2 ) 8
iR-1~)
.
(5.21)
-1 Proof. Put F = U
3 f.
For each 0. E IR R, c
- 76 -
f(o.) exp(- 1I~1I2). J e(o.·x)F(x)d~(x) .
IRIN
The functions on both sides of the equality sign can be extended to analytic
functions.
So for all 0.,13 E IR R, c
We get for all 0.,13 E IR R. c
J e(o.+i13·x)F(x)d\.l{x) = exP(1I0.1I2 -1I1311~ +2i(a..S»)[U
1fJ(0. + 1(3) •
IRIN
Fix ~ E IRR. • Then we have c
[T(R,~)fJ(r;) = [U3
(T(R,f;)F)](r;)
:: e-II~1I2/2 J e(l;;·x)e-i(f;.x)F(R-1x)d~(x)
IRIN
Substitution of y = R-1x and use of (5.21) yield:
Hence (5.19) is proved.
For x E IR IN we have
Fix I;; E IRR. . Then c
= e _II ~1I2 /2
2 -1 exp(-(IIf;1I + 2(~.x)}/4)F(R (x +~» .
(5.22)
(5.23)
Since dJ.,l (x + t;;)
e
- 77 -
-lIsIl2/2 f exp«l;;.x) - (111;112
+ 2(I;.x»/4) •
mlN
-1 • F(R (x + t;;»dl1(x) •
2 = exp(-(IIF;II + 2{t;;.x»/2)dl1(x) we get
exp(- 1Ir,1I2
- (t;;.r,;) _ ~) 2 4 J
J
-1 -1 I; (x,R r,+R -) 2
e F(x)dl1(x)
2 exp(- (F;;l;;) - II ~" ) f(R- 1
(I;; + t» So (5.20) is proved. o
Because of the previous result we define for all (R,~) E o(m~2) x m£2 the "" ~
operators T(R/~) and S(R,I;) from Minto M by
II r,1I 2 _ -1 1 exp(- 2 i(l;.~»[U1f](R l; - iR- 1;)
[S(R,s)fJ(l;) = exp(- (F;;l;) - 11~1I2)f(R-1(l; + t», l; E m£2' f ( M •
With the aid of the reproducing kernel of M it can be shown just as for
o{m£2) that T and S are unitary representations of the group o(m£2) x m~2
in M. We note that these representations are also irreducible. See the re
mark before theorem 5.20.
We make two more remarks on Bargmann's work [BA 1] and [BA 2].
(5.24) The operator U1U
3 is a unitary isomorphism from £2 onto B. In [Ba 1], sec
tion 2 a unitary isomorphism A from (mq) onto the Bargmann space Bq is
q introduced by
[Aqf](Z) = q/4 f exp(-~[(z.z) + lIun 2] + /2(z.u»f(u)du ,
11
- 78 -
Let [ ]q denote the space {F E £2 I 3 gEL
2 (IR q ,exp (-II ull 2 /2) du)
V IN: F(x} = g({x.e 1),···,(x.eq»} • XEIR
Then for F E [£2Jq with F(x) = g«x.e1
) , ••• ,(e.eq
» for all x E mlN we have
(5.25) In [Ba 2J, section 3, it has been suggested that representations of the kind
(5.22) and (5.23) can be introduced.
- 79 -
6. Characterization of M and related spaces
In this section we characterize the elements of the spaces M , M and B. n
For a certain class of positive self-adjoint operators C in M it appears that
the elements of SM,C and TM,C are real analytic functions on TIR~2'A and
SIRt2
,A respectively. Here A denotes a positive self-adjoint operator, which,
in our terminology, is exponentially coupled with C on IRt2 x M.
For a certain class of positive self-adjoint operators C' in B similar re
sults have been derived for the elements of SB,C' and TB,C" A subset of the
latter mentioned class admits the characterization of the elements of the
spaces SB,C' and TB,C' as analytic functions with a simple growth condition.
Our first result concerns the characterization of the elements in M . n
Theorem 6.1. Let n E IN and let f be a function from IRt2 into ~.
The following statements are equivalent:
(a) f EM. n
(b) The function f is continuous. The functions fiq: IRt2 +~, q E IN,
defined by
a
satisfy
(i) sup{1I fiqU I q E IN} < 00.
~ (H) V f;q(y) -+ f(y), (q + 00).
YEIR£2
E Y q
Proof. (a) .. (b) :
f;q = pqf we have n
Let f EM. Because n
of theorem 3.6 f is continuous. Since
II f II :s II fll M' q~
Since lim P~f = f in M-sense we have for all y E IR £2 lim f;q(y) = fey).
(b)
q+oo ~ (al: For each q
q+oo E IN the function f iq restricted to IRq is harmonic
and II fiC!t1 ~ = II fiC!t1 W
(6.2)
- 80 -
Put M = sup{11 f;qU M I q E IN}.
The sequence (f;q) is norm bounded in M • So it has a weak convergent sub-• q n ,q.
sequence (f J). with limit hEM • J n
Fix y E IR~2' Then
u.;t)n) _ ;q. n h (y) ::: (h, lim (f ) ( •• y) )
n~ M-j~
n~ M
;q. = lim f ) (y) = fey) .
j+oo
So f = h E M . n
In a similar way the elements of M can be characterized.
Theorem 6.2. Let f be a function from IR~2 into ~.
The following statements are equivalent:
(a) f E M.
0
(b) There exist continuous functions fn: IR~2 +~, n ~ 0, such that for
all A E IR and all x E IR R.2
00
f (Ax) I n=O
The holomorphic extension A + f(Ax), A E IR, will be denoted by
f ( • 1 x) •
The functions f;q: IRR.2 + ~ defined by
J IAI=r
satisfy
f q-l
S
(i) sup{1I f;~1 I q E IN} < 00.
~ (ii) V f;q(y) + fey}, q + 00.
YEIRR.2
(Note that f (vx) = vnf (x). If f E M then the expressions on the right n n n
side of (6.2) and (6.3) are identical.)
(6.3)
Proof. (b) ... (a): Fix y E nu2
.
From lemma 2.6 it follows that
1 lim ---=---
21Tia 1 r--' q-
For all n E IN u {a} put
a q-1
- 81 -
f f IAI=r q-1
S
00
L m,n=O
E Y • P~(~. IIEqyll) Jdaq_1 (~)dA
q
Since both series in (6.3) are uniformly convergent on Co ,r q-1 x S we get
00
L n=O
fiq(y) n
So for all q E IN the function f iq is well defined and its restriction to
]Rq is harmonic, hence f iq E ~~. The remaining part of the proof consists of
the same arguments as used in the proof of (b) ... (a) in theorem 6.1.
(a) ... (b): By assumption we have for each A E IR and x E IR~2
00
f(h) I n=O
A nCp fJ (x) n
Also, the functions P f are continuous. n
Apparently for all q E IN
The remaining part is trivial. o
Next we characterize the elements of M , M and B in a different way. The ren
producing kernel is a very important tool in this characterization.
Theorem 6.3. Let n E IN and let f denote a function from IR~2 into ~.
Then f E M iff there exists c > 0 sich that for each ~ E IN and all n
a j E ~, Yj E IRi2 with 1 ~ j ~ i
J/,
I L j=l
u.f(""Y.T1 J J
- 82 -
1 n Proof. The function (x,y) -+ -. (x.y) is the reproducing kernel of M • Apply --- n. n theorem 1.2.
Theorem 6.4. Let f denote a function from IRJ/,2 into ~.
Then f E M iff there exists c > 0 such that for each J/, E IN and all
uj
E ¢, Yj
E IRJ/,2 with 1 S j S J/,
J/,
I I j=l
u. f (Y .) I J J
Proof. The function (x,y) -+ exp(x.y) is the reproducing kernel of M. Apply
theorem 1.2.
Theorem 6.5. Let f be a function from ~J/,2 into ¢.
Then f ~ B iff there exists c > 0 such that for each J/, E IN and all
uj
E ~, Wj E (J/,2 with 1 S j S J/,
Proof. Similar to the proof of theorem 6.4.
In theorem 3.20 we stated that the functions I.P , sEMI, belong to M. Here s
o
o
o
we give another proof based on theorem 6.4. To this end we need the auxilia-
ry statement:
00
(6.4) Let the series I j=l
y. be absolutely convergent. Then J
L tEMI
m
1 t!
IT j=l
t. y.]
J
Now fix SEMI. We choose J/, E IN, a i < a, Yi E IRJ/,2 with 1 S i $ t. Then
~
I I i=l
CUP (y.) 12 :s 1. s 1.
=
=
- 83 -
I I n=O tEM1
n
R-
L (li(lk k,i=1
oo
I n=O
00
00
I 1 IT
t€MI t j=1 n
!l
L I 1 n -. (Yk· Y')
k,i=1 (li(lk
n=O n. 1.
!l I (li(lk exp(yk·yi )
k,i=1
=
[(Yi)j(Yk)j]
See (6.4)
So for the constant c in theorem 6.4 we may take the value 1, whence tp E M. s
In the second part of this section we characterize some analyticity and tra
jectory spaces.
First we repeat the introduction of these spaces.
Let Y denote a Hilbert space and let B denote a positive self-adjoint opera
tor in Y.
The analyticity space Sy,B and the trajectory space Ty,B are defined by
I tB {f E Y 3
t>O: f E D(e )}
and
-tB For F E Ty,B and t > 0 the element F(t) will also be denoted by e F.
Let B+(IR+) denote the set of all everywhere finite Borel functions W on
IR (= [0,(0» with the properties +
and
The topology on S B is the locally convex topology induced by the seminorms Y,
P,I.' ~ E B (IR ), defined by 'I' + +
See [EG], Definition 1.1.1 and lemma 1.1.4.
- 84 -
The topology on Ty,B is the locally convex topology induced by the semi norms
0t' t > 0, defined by
Definition 6.6. Let F denote a Hilbert space of functions on a Hilbert space
X. Let A and C denote positive self-adjoint operators in X and F, res
pectively, such that for each t> 0
X E X, f E F .
In our terminology the pair [A,C] is said to be exponentially coupled
on X x F.
Definition 6.7. Let [A,C] denote an exponentially coupled pair on X x F. For each f E SF,C we define f: TX,A ~ ~ by
feu) [eTCf] (e-TAu)
TC with T > 0 so small that f E D(e ), u ETA' X,
(The definition of f does not depend on T since [A,C] is an exponential-
ly coupled pair.) -For each F E TF,C we define F: SX,A ~ t by
TA with T > 0 so small that v E D(e ), v E SX,A' (The definition does not
depend on T.) Let
and let
We give some examples of function spaces which appear as SF,C- and TF,cspaces.
In definition 2.13 the Hilbert space Mr has been introduced. From theorem
2.22 it follows that Mr consists of all functions f on IRq which are harmo-
nic and satisfy
- 85 -
The operator Qq is the orthogonal projection from ~ onto ~. n n
In ~ the positive self-adjoint operator C is defined by q
D(C ) = {f E ~ I q
C f q I
n=O f E D(C )
q
We remark that for each t > 0, X E IRq and f E ~
-te [e qfJ (x)
-t fee x) •
So [I ,e ] is an exponentially coupled pair on IRq x ~. q q
Now S =:. T = IRq with usual identifications, so the elements of
S IRq,I
q. IRq I , q
and T are functions from IRq into ~. ~,e
q ~,e
q
Lemma 6.8. Let f E ~.
Then for each r > 1 there exists c > 0 such that for each x E IRq
2 I f (x) I ~ c exp (r":" )
Proof. Take r > 1 and x E IRq. Using (2.7) we get
00
I f (x) I ~ L n=O
00
Kq(x,X»~ = ~ II flM ( I n n=O (Corollary 2.17)
00 r (s.) dq
= II filM ( I 2 n II xii 2n) ~ n=O 2
nr (n + %)
00 r(%)d~n~ rnll xll2n) ~ = II filM ( I
r(n + ~)r n 2n .n! n=O
2
Let
M r
- 86 -
O} •
Clearly M <~. So we get r
Theorem 6.9. Let g denote a function from IRq into ~.
Then the following statements are valid:
(al g E S iff g is harmonic and Jfl,c
q
3 3 V r,O<r<l c>O q
2
I ( ) I (rll xii ) g x $ c exp ---4---XElR
(b) g € t iff g is harmonic and Jfl,c
q
V 3 V r>l c>O XElR q
2
I ( ) I ( rll xii ) g x $ c exp ---4---
~
Proof. (a) ~: Let f € S with f g. So g is harmonic. Further there Jfl,c
q tC exists t > 0 such that f E D(e q). For all x E lR
q we have
tC t g(x) = f(x) = [e qf](e- x}
Because of lemma 6.8 there exists c > 0 such that for all x E IRq
Take r
tC -t 2 2 I [e qf] (e-tx) I ( t II e xII ) ( -t ~) $ c exp e 4 = c exp e 4 .
-t e
So for each x € lRq
2 Ig(x) I $ c exp(rll:" ) •
(a) ~: Let r,c € IR with 0 < r < 1 and c > 0 be such that for all x € lRq
2 Ig(x) I :$; c exp(rll:" )
(6.5)
(6.6 )
- 87 -
Choose t > 0 such. that re 2t < 1.
Then the harmonic functions· q and 9 (e t.) have a finite MI-norm. tC
Since q(et~) :: e q we see 9 E S • M!,c
q (b) -: Let f E T
WI'C q
with f ::: g. Since e -t .a
gee .) E M-" we see that
q is harmonic on m q . 2t
Fix r > 1. Choose r' > 1,. t > 0 such that r ::: r'e .. Because of lemma 6.8
there exists c > 0 such that for all x E mq
I 9 (x) I "" I -t t I gee (e x» $
t 2 $ ( • II e xii ) c exp r 4
$ IIxU
2 c exp(r -4-)
-.c -t • .0, -(t+T)
(b) *": Clearly for all t > 0 9 (e .) E M -. Sl.nce g (e .) ::: q -t
e 9 (e .) -t
for t, T > a we get t -+ g (e .), t > 0 € T MI,c
q
and hence gET AfI,c
q
The second example concerns the Bargmann space B of infinite order.
Le t ( it ,) , be a sequence in m such that it, > O. In the Bargmann space B we J J J
have the orthonormal basis {u I SEMI} where s
us(z)
Let A denote the
Ae, = J
Let C denote the
Cu s
00
1 II = --
1ST j=l
positive
A ,e, ] ]
,
positive
(z. e ,) J
Sj
self-adjoint operator
j E IN .
self-adjoint operator
in 4:£2 with
in B with
It is easy to see that for all t > 0, sEMI and z € 4:t2
[e -tCu ] (z) = u (e -tAz ) • s s
Hence for all t > 0, f € Band Z E <tt2
o
- 88 -
-tC -tA [e f](z) = fee z).
SO [A,C] is an exponentially coupled pair on ~t2 x S.
Lemma 6.10. Let [A,C] denote an exponentially coupled pair on ~t2 x S defined
as in (6.5) and (6.6).
The following statements are valid:
(a) If f E SS, C' then f is continuous on T~t2'A' (b) If f E S, then f is continuous on ~t2 •
(c) If f E TS,C' then f is continuous on S£t2,A'
Proof. (b): Fix f E S.
The mapping w + exp(o.w) from ~t2 into S is continuous.
Since for each u,v E ~t2
If(u) - f(v) I :::; IIfIlS.llexp(o.u) - exp(o.v)II S
we conclude that f is continuous on S. tC
(a): Fix f € SS,C' There exists t > 0 such that f € D(e ).
Fix u E T~t2'A and £ > O.
Because of (b) there exists 0 > 0 such that
Then for all v € T~t2'A with
we get the estimate
(c): Fix f € TS,C'
As in [EG], theorem 1.2.12, we prove that there exists \.p E B+(lR+) and
h E S such that f = \'p(C)h.
In addition \.p satisfies \.P(Sl + S2) s \.P(Sl)·~(S2)' SI,S2 € lR+. Let
M = lIe-(1/n)Cfll . n
We choose d 1 = -1 and for n ~ 2 d such that n
and
n - (l/n) C (C) II e Xed ,00)
n
d > d + 1 n n-1
Put
00
- 89 -
2 M1
<- nM n
<.p( A) \' 2A!n L e Xed d ) (A)
n=1 n' n+1
00
h L n=l
Then h E Band f = <.p(C)h.
For Sl,S2 ~ 0 we have
and in general for £ E IN, Sl, •• "St > 0
j=l IT <.p(S.)
J
A ~ 0 •
For all a,b E S we derive an estimate with the aid of property (1.8): m £2,1\
2 II <.p(C) [exp(o.a) - exp(o.b)JIIB
L [u (a) - u (b) Ju II B2 S S S
SEMI
00
I [<.p( Y. A.s.)J2
Iu (a) - u (b) 12 :::; J J S S SEMI j=l
00
L IT SEMI j=l
L SEMI
00
IT j=l
2s. [<.p()..)] J
J u (a) - u (b) 12
S S
L lu (<.p(A)a) - u (<.p(A)b) 12 S S
sEMI
- 90 -
= II I 2 (u (q:>(A) a) - u (q:>(A)b»u IIS=
s s s SEMI
2 lIexp(o.q:>(A)a) - exp(o.!.f)(A)b)II S
Fix a E Sand E > O. c::.t2
,A
Let 8 > a be such that
v I . lIexp(o.!.f)(A)a) - exp(o,z)II S ZEltt2
,II Z-!.f)(A) a 1<8'
For all b E S n A with 1I!.f)(A)a - !.f)(A)bll = P (a - b) < 8 we have the estimate ~~2' !.f)
If(a) - feb) I = II (!.f)(C)h, exp(o.a) - exp(o.b»sl :s
::;; IIhll S.II!.f)(C) [exp ( •• a) - exp{· .b) ]11 S =
:s; IIhIiS.llexp(o.!.f)(A)a) - exp(o.!.f)(A)bJIl S :s; E • o
Lemma 6.11. Let [A,C] denote an exponentially coupled pair on Itt2 x S defined
as in (6.5) and (6.6).
The following statements are valid:
(a) If f E SS,C' then f is analytic on TCt2 ,A'
(b) If f E S, then f is analytic on 1tR. 2 •
(c) If f E TS,C' then f is analytic on S\tR.2
,A'
Proof. (b): Fix f E S and fix a,b E ~t2' "" ,...,
The function A + feE a + AE b) is analytic on ~. Since q q
feE a + AE b) = [T f](a + \b) q q q
and since
the sequence (A + [T f](a + Ab), A E c::.} converges to A + f(a + Ab), A E ~ q q
uniformly on compacta in ~. Hence A + f(a + Ab) is an analytic function on ~.
Because of the continuity of f, the function f is analytic on ~t2' tC
(a): Fix f € SS,C' Let t > 0 such that f € D(e ). Let u,v € T~t2,A'
For A E ~ we have
- 91 -
Apply (b) and lemma 6.9.
(c): Fix f E TB ,. Let a,b E S~" A ,G ~~2,JI
til and let t > 0 be such that a,b E Dee ).
For A E ([ we have
Apply (b) and lemma 6.9.
Lemma 6.12. Let (A.>. be a sequence in IR such that A. > 0 and for all t > 0 J J ]
I e -2tA
j < (XI
1
Let A denote a positive self-adjoint operator in ~9.,2 with Ae. = A.e .• ] J ]
Let f: C9.,2 ~ ([ be an analytic function such that there exist t,c > 0
with
-tA 2 -; c exp(lle zll /2) .
Then f € B.
Proof. Because of theorem 1.15, Hartogs' theorem, the function fl is ana-ceq
lytic on a::q
. Put
Cl. = lim sup q~
If Cl. < (XI then fEB. We have
Ct ~ lim sup q~
= lim sup q~
= lim sup q~
2 c
2 c
2 % lim sup c
q-)o<XI
f exPC_ ~ (1 - e
nq j=l
([q
q IT
j=l
q IT
j=l
1
f exp (- (1 -1T
(t
1
(1 - e
-2tA. 2 J) 1 z.1 ) dz
]
-2tA. 2 J) 1 z.1 ) dz. e
2 c
00
IT j=l
] ]
__ --.:1=---__ < 00
-2tA j) (1 - e
(6.7)
because
00
L j=l
e -2tA
j
- 92 -
We have the following characterization.
Theorem 6.13. Let [A,C] denote an exponentially coupled pair on ~i2 x B,
o
where A and C are defined as in (6.5) and (6.6). Moreover let (Aj)j C IR+
have the property that for each t > 0 the series
00 -2tA .
L e J
j=l
is convergent. The following statements are valid: -
(a) f E SB,C iff f is an analytic function from TCi2
,A into ~ and
-tA 2 ~ c exp(lIe zll /2) .
-(b) f E TB,C iff f is an analytic function from S into ~ and Ci
2, A
tA 2 ~ c exp(lle zll /2) .
Proof. The proof is based on the lemmas 6.10, 6.11 and 6.12 and on the tech-
niques used in the proof of theorem 6.9. o
Finally we give characterizations of a class of analyticity and trajectory
spaces related to M. The same results can be fbrmulated for the Bargmann
space B instead of M. We remark that these characterizations lean upon theo
rem 6.4, i.e. the characterization of M with the aid of its reproducing
kernel.
Theorem 6.14. Let [A,C] denote an exponentially coupled pair on IRi2 x M. Let f denote a function from TIRi A into ~.
2' Then f E SM,C iff for some t > 0 there exists c > 0 such that for each
i E IN and all u. E ~, YJ' E T n with 1 ~ j ~ i J lRx..
2,A
i
I L j=l
u.f"('Y.T1 ~ c( J J
- 93 -
Proof. -: Let
-tA I Span ( {exp ( •. e y) y € T m t A} ) 2'
We define m: Wt
-~ It by
t m( I
j=l
-tA a.exp(o.e y.» ] ]
From (6.7) it follows that for all w E Wt
m (w) ~ ell wil •
Since Wt is dense in M there exists 9 € M such that for all w E Wt
mew) = (w,g)M •
For each y E mt2
we have
-tA fey) '" m(exp( o.e y»
-tA ::; (exp (· .. e y) ,q) A{
-tC So f '" e g.
-tA 9 (e y).
Hence f E SM,C. ~ - tC
~: Let h E SM,C with h = f. Then h E D(e ) for some t > O.
Choose Q, E IN, a. (: It, y. E Tm " wi th 1 ::; j ~ to Then we have J ] ~2,A
t
L 1
tC -tA I a . [e hJ (e y .) ] ]
Take c o
(6.8)
- 94
Theorem 6.15. Let [A, C] denote an exponentially coupled pair on m t2 x M.
Let f denote a function from Sm 9, A into (J;.
2' Then f E fM,C iff for each t > 0 there exists c > 0 such that for each
t E :IN and a 11 a j E <I!, Y j Em! 2 wi th 1 ::;:; j ~ !
t I I j=l
! a.f(e-tAy.) I ::;:; c( I
J J k,
Proof. ~: Let
and let
W = Span({exp(o.y) lYE IR!2}) .
Define m: Ws ~ <I! by
Fix t > O.
t m( I
j=l a. . exp ( ... y . » =
J J
Because of (6.8) we get for each! E :IN and all aj
E <I!, Yj E mt2 with
1 ~ j :5 !
I -tC! !
moe (l. ajexp(o.y.» I s; ell I a.exp(o.y.)IIM• 1 J j=l J J
So we can extend m 0 e-tC to the whole of M. Hence there exists gt E M such
that for all w E W
-tC moe (w)
For all Y E mt2
we have
f (e -tAy) -tA -tC m(exp(".e y» = moe (exp(·.y» =
Whence we conclude f(e- tA .) E M. SO t ~ f(e- tA.), t > 0 E TM,C or equivalent
ly f E IM,C'
- 95 -
·:Letg E TM,C with 9 = f. Let t > O. Choose 9.. E IN, aj
E Q:, Yj
E IR9..2 with
1 ~ j ~ 9... Then we get
9.. tA I l: (l . f (e - Y') I j=l J J
9..
I l: j=l
-tC tA -tA I a.[e gJ(e e y.) = J J
I ~ -tC I (L a . exp ( •. y .) , e g) M ~ 1 J J
-tC We take c = II e gll M • o
Moreover theorem 6.13 and 6.14 can also be formulated for the Bargmann space
B in exactly the same way.
- 96 -
Acknowledgement
I wish to thank S.J.L. van Eijndhoven and J. de Graaf for inspiring discus
sions and helpful remarks on the subjects in this manuscript.
- 97 -
References
[Ar] Aronszajn, N., Theory of reproducing kernels, Trans. A.M.S., 68, 337-404
(1950) •
[Ba 1J Bargmann, V., On a Hilbert Space of Analytic Functions and an Associated In
tegral Transform, Part I, Commun. Pure and Appl. Math., Vol. XIV, 187-214
(1960) .
[Ba 2J Bargmann, V., On a Hilbert Space of Analytic Functions and an Associated In
tegral Transform, Part II, Commun. Pure and Appl. Math., Vol. xx, 1-101 (1967).
[Ba 3J Bargmann, V., Remarks on a Hilbert Space of Analytic Functions, Proc. N.A.S.,
Vol. 48, 199-204 (1962).
[EG] Eijndhoven, S.J.L. van, Graaf, J. de, Trajectory Spaces, Generalized Func
tions and Unbounded Operators, Lecture Notes in Mathematics 1162, Springer,
1985.
[Gr] Graaf, J. de, Papers dedicated to J.J.Seidel, EUT Report 84-WSK-03, 166-182.
[Gui] Guichardet, A., Symmetric Hilbert Spaces and Related Topics, Lecture Notes
in Mathematics 261, Springer, 1972.
[Hi] Hida, T., Brownian Motion, Applications of Mathematics 11, Springer, 1980.
[Ho] Hormander, L., An introduction to complex analysis in several variables, nd
2 ed., North Holland Publ. Comp., 1973.
[KoJ Kono, N., Special functions connected with representations of the infinite
dimensional motion group, J. Math. Kyoto Univ., 6-1, 61-83 (1966).
[MOS] Magnus, W., Oberhettinger, F., Soni, R.P., Formulas and Theorems for the
Special Functions of Mathematical Physics, Die Grundlehren der mathematischen
Wissenschaften in Einzeldarstellungen, Band 52, 1966.
[MU] Muller, C., Spherical Harmonics, Lecture Notes in Mathematics 17, Springer,
1966.
[Or] Orihara, A., Hermite polynomials and infinite dimensional motion group, J.
Math. Kyoto Univ., 6-1, 1-12 (1966).
[SO] Schempp, W., Oreseler, B., EinfUhrung in die harmonische Analyse, Mathema
tische LeitfAden, B.G. TeUbner, Stuttgard, 1980.
- 98 -
ESe] Segal, I.E., The complex wave representation of the free boson field, Topics
in Functional Analysis, Advances in mathematics supplementary studies, Vol.3,
321-343, Academic Press, New York, 1978.
[Tr] Treves, F., Topological Vector Spaces, Distributions and Kernels, Academic
Press, New York, 1967.
[Vi] Vilenkin, N.J., Special Functions and the Theory of Group Representations,
Translation of Mathematical Monographs, Vol. 22, A.M.S., 1968.
[Yos] Yosida, K., Functional Analysis, Grundlehren der mathematischen Wissenschaf
ten 123, Sixth edition, Springer, 1980.