the hypergeometric approach to integral transforms and convolutions

Mathematics and Its Applications
Volume 287
by
Yurii F. Luchko Department of Mathematics and Mechanics, Beylorussian State University, Minsk, Byelorussia
SPRINGER-SCIENCE+BUSINESS MEDIA, B. V.
Library of Congress Cata1oging-in-Pub1ication Data
Yakubovich. S. B. (Semen B. J. The hypergeometric approach to lntegral transforms and
convolutlons ! by Semen B. YakuboV1Ch and Yuri 1 F. Luchko. p. cm. -- (Mathematics and ltS appllCatlOnS ; v. 2871
Includes bibllographlcal references and lndexes. 1. Integral transforms. 2. Hypergeometrlc functions.
3. Convolutions (Mathematlcs) 1. Luchko. Yurll F. II. Title. III. SerJes- Mathematics and ltS applicatlOns (Kluwer Academic Publ ishers) ; v. 287. QA432.Y35 1994 515'.723--dc20
ISBN 978-94-010-4523-0 ISBN 978-94-011-1196-6 (eBook) DOI 10.1007/978-94-011-1196-6
Printed on acid-free paper
All Rights Reserved © 1994 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1994 Softcover reprint ofthe hardcover Ist edition 1994
94-14888
No part of the material protected by this copyright notice may be reproduced or utilized in any form Of by any means, electronic or mechanica1, including photocopying, recording or by any information storage and retrieva1 system, without written permission from the copyright owner.
Contents
Preface .... ix
1 Preliminaries 1 1.1 Some special functions 1 1.2 Integral transforms . . 8
2 Mellin Convolution Type Transforms With Arbitrary Kernels 15 2.1 General Fourier kernels. . . . . 15 2.2 Examples of the Fourier kernels 18 2.3 Watson type kernels 25 2.4 Bilateral Watson transforms . . 30 2.5 Multidimensional Watson transforms 34
3 H- and G-transforms 41 3.1 Mellin convolution type transform with Fox's H-function as a kernel 41 3.2 Mellin convolution type transforms with Meijer's G-function as a kernel 50 3.3 The Erdelyi-Kober fractional integration operators. . . . . . . . . .. 54
4 The Generalized H- and G-transforms 59 4.1 The generalized H-transform . . . . . . . . . . . . . . . . . 59 4.2 The generalized G-transform . . . . . . . . . . . . . . . . . 62 4.3 Composition structure of generalized H- and G-transforms 64
5 The Generating Operators of Generalized H-transforms 69 5.1 Generating operators in the space ~M;'~ 69 5.2 Examples of the generating operators 75
6 The Kontorovich-Lebedev Transform 79 6.1 The Kontorovich-Lebedev transform: notion, existence and inversion
theorems in M;'~(L) spaces .. . . . . . . . . . . . . . . . 79 6.2 The Kontorovich-Lebedev transform in weighted L-spaces . 85 6.3 The Kontorovich-Lebedev transform in weighted L 2 spaces 94 6.4 The Kontorovich-Lebedev transform of distributions. 99 6.5 The Kontorovich-Lebedev transform in L,,-spaces . 103
v
7 General W-transform and its Particular Cases 109 7.1 General G-transform with respect to an index of the Kontorovich-
Lebedev type . . . . . . . . . . . . . . . . . . . . . . . . . . 109 7.2 General W-transform and its composition structure . . . . . 118 7.3 Some particular cases of W-transform and their properties . 120 7.4 F3-transform.......................... . 126 7.5 L2-theory of the Kontorovich-Lebedev type index transforms . 130
8 Composition Theorems of Plancherel Type for Index Transforms 139 8.1 Compositions with symmetric weight . . . . . . . . . . . . . . . 139 8.2 Compositions with non-symmetric weight. . . . . . . . . . . . . 143 8.3 Constructions of index transforms in terms of Mellin integrals . 145
9 Some Examples of Index Transforms and Their New Properties 149 9.1 The Kontorovich-Lebedev like composition transforms. . 149 9.2 Some index transforms with symmetric kernels. . 153 9.3 The ~- and ~- index transforms . . . . . . . . . 156
10 Applications to Evaluation of Index Integrals 167 10.1 Some useful representations and identities . 168 10.2 Some general index integrals . . . . . . . . . . 171
11 Convolutions of Generalized H-transforms 173 11.1 H-convolutions . 173 11.2 Examples of H-convolutions . . . . . . . . . . 178
12 Generalization of the Notion of Convolution 183 12.1 Generalized H-convolutions . . 183 12.2 Generalized G-convolutions . 187
13 Leibniz Rules and Their Integral Analogues 189 13.1 General Leibniz rules . . . . . . . . . . . . . . . . . . . . . . . .. . 190 13.2 Modified Leibniz rule . . . . . . . . . . . . . . . . . . . . . . . .. . 193 13.3 Leibniz rule for the Erdelyi-Kober fractional differential operator. . 195 13.4 Modification of the Leibniz rule for the Erdelyi-Kober fractional dif-
ferential operator . . . . . . . . . . . 198 13.5 Integral analogues of Leibniz rules . . . . 202
14 Convolutions of Generating Operators 14.1 Convolutions in the Dimovski sense. General results 14.2 Examples of convolutions in the Dimovski sense ..
205 .. 205 . . 210
15 Convolution of the Kontorovich-Lebedev Transform 213 15.1 Definition and some properties of a convolution for the Kontorovich
Lebedev transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 15.2 The basic property of convolution. Analogues with the Parseval equality218
VI
15.3 On the inversion of the Kontorovich-Lebedev transform in the ring La 221 15.4 The space La as the commutative normed ring of functions with ex
ponential growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
16 Convolutions of the General Index Transforms 229 16.1 Convolutions of the Kontorovich-Lebedev type transforms 229 16.2 The convolutions for the Mehler-Fock and the Lebedev-Skalskaya
transforms. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . 236 16.3 The convolution of the Wimp-Yakubovich type index transform ... 238
17 Applications of the Kontorovich-Lebedev type Convolutions to In- tegral Equations 241 17.1 Kontorovich-Lebedev convolution equations of the second kind . 241 17.2 General composition convolution equations . 246 17.3 Some results on the homogeneous equation . 247
18 Convolutional Ring Ga 253 18.1 Multiple Erdelyi-Kober fractional integrodifferential operators . 254 18.2 Convolutional ring Ga ••.•.••••••••••••••••• • 261
19 The Fields of the Convolution Quotients 265
19.1 Extension of the ring (Ga ,;, +) . 265 19.2 Extension of the ring (La, *, +) .... . . 272
20 The Cauchy Problem for Erdelyi-Kober Operators 277 20.1 General scheme . . . . . . . . . . . . . . . . 277 20.2 Differential equations of fractional order . . 279 20.3 Differential equations of hyper-Bessel type . 282
21 Operational Method of Solution of some Convolution Equations 287 21.1 Integral equations of Volterra type 287 21.2 Integral equations of second kind with Kontorovich-Lebedev convolution292
References
Notations
vii
295
311
317
321
Preface
The aim of this book is to develop a new approach which we called the hyper geometric one to the theory of various integral transforms, convolutions, and their applications to solutions of integro-differential equations, operational calculus, and evaluation of integrals. We hope that this simple approach, which will be explained below, allows students, post graduates in mathematics, physicists and technicians, and serious mathematicians and researchers to find in this book new interesting results in the theory of integral transforms, special functions, and convolutions.
The idea of this approach can be found in various papers of many authors, but systematic discussion and development is realized in this book for the first time.
Let us explain briefly the basic points of this approach. As it is known, in the theory of special functions and its applications, the hypergeometric functions play the main role. Besides known elementary functions, this class includes the Gauss's, Bessel's, Kummer's, functions et c. In general case, the hypergeometric functions are defined as a linear combinations of the Mellin-Barnes integrals. These ques tions are extensively discussed in Chapter 1. Moreover, the Mellin-Barnes type integrals can be understood as an inversion Mellin transform from the quotient of products of Euler's gamma-functions. Thus we are led to the general construc tions like the Meijer's G-function and the Fox's H-function. In Chapter 1 we give some preliminary notions of the theory of functions of hypergeometric type, their asymptotic behaviour, integral representations and expressions through the hyper geometric series. Moreover, we observe that the considered special functions have different asymptotic behaviour, but the asymptotic of the integrand in their Mellin Barnes representations is based only on the asymptotic formula of gamma-function. This property will be used in consideration of the integral transforms in the special space of functions. In Chapter 1 we present various classes of integral transforms and give the elements of the theory of the Mellin transform and convolution. The hypergeometric approach allows us to consider all these integral transforms from the same point of view by means of the Parseval equality for the Mellin transform and moreover to obtain new constructions of integral transforms and convolutions.
Chapter 2 deals with the theory of the Mellin convolution type transforms with general so-called Watson and Fourier kernels. The L t - and L 2-theorems for these transforms and the inversion theorems in special space M-t(L), which is isomorphic to the space Lt(R), are established. The classical examples such as the sine-, cosine-
ix
Fourier transforms, the Hankel transform, and some new ones are given. Moreover these general transforms on positive half-axis are generalized to the bilateral case on real axis and become the known Hartley transform which gives the essential multidimensional analogues of the Watson transforms.
In Chapter 3 we consider the most general particular cases of the Mellin con volution type transforms with the G- and H-functions as kernels (so-called G- and H-transforms). These transforms involve, as particular cases, all known convolu tion transforms and the existence and inversion theorems for them are established. Moreover, we indicate such important particular cases as the Erdelyi-Kober frac tional integration operators, which are also discussed in the following chapters.
Chapters 4 and 5 are devoted to construction of the generalized G- and H transforms by means of the Mellin-Parseval equality in the special functional space M;:~(L) and their so-called generating operators. It is shown that the main role in their composition structure is played by the direct and inverse modified Laplace transforms. In Chapter 5, the generating operators for Fourier, Stieltjes, Borel Dzrbasjan, Obrechkoff integral transforms are constructed.
The following Chapters 6-10 deal with the index transforms, which are principally different from the Mellin convolution type transforms, but can be also investigated by means of the hypergeometric approach. The notion of index transform based on the special kind of integration in the inversion formula, namely with respect to index (parameter) of special function involving in the kernel of this transform. First we discuss both the known index transforms as the Kontorovich - Lebedev trans form, the Mehler-Fock transform, the Olevskii transform, the Lebedev-Skalskaya transforms and the new ones with cylindrical functions as kernels and more general constructions. In Chapter 6 we give some new theorems on the Kontorovich-Lebedev transform including the Lp-case and the space of distributions. In Chapter 7, fol lowing to J.Wimp, we generalize the mentioned classical index transforms on the case of Meijer's G- and Fox's H-functions with the distinguished indices of integra tion. In Chapter 8, we consider the general compositions of the Watson transforms and we construct the respective index pairs of integral transforms. Chapter 9 deals with the interesting examples of index transforms and convolutions including the Lebedev-Skalskaya !R- and ~-transformsrelated to the Kontorovich- Lebedev trans forms. Here, the Lp-theorems for the Lebedev-Skalskaya transforms are established. In Chapter 10 we remark about some ways of evaluation of the index integrals of the Kontorovich-Lebedev type and, more generaJly, with the Meijer's G- and Fox's H-functions as the kernels.
In the following Chapters 11-14, we return to the Mellin convolution type trans forms to construct the respective convolutions of them. The idea of convolution constructions of the general convolutions by means the double Mellin-Barnes inte grals was realized by the first author and is developed in this book. We construct the convolutions for the generalized H- and G-transforms in the Dimovski's sense and give some interesting examples including the convolutions for Erdelyi-Kober fractional integro-differentiation operators and the convolutions for the generating operators.
Chapters 15-16 deal with the convolutions for the index transforms. First we give
x
the theory of convolution for the Kontorovich-Lebedev transform from the analytic and algebraic points of view. In Chapter 17, the applications to the solution of new type of convolution equations of second kind with symmetrical kernels are considered.
In Chapters 18-19, the operational calculi for the multiple Erdelyi-Kober integro differential operators and for the Kontorovich- Lebedev convolution are constructed. For this the convolution rings of functions continuous and summable with special weight are extended to the fields of convolution quotients.
Last Chapters 20-21 deal with the applications of the operational method to the solution the Cauchy problems for the equations containing the multiple Erdelyi Kober derivative with constant coefficients, for the hyper-Bessel differential equation and differential equation of fractional order. Moreover, some class~s of integral con volution equations of second kind are considered, namely some equations of Volterra type and, in additional to Chapter 17, the second kind integral equations with the Kontorovich-Lebedev convolution.
For the sake of convenience, we give author, subject and notation indexes at the end of the book.
This book is written primarily for teachers, researchers and graduate students in the areas of special functions and integral transforms. Research workers and other users of special functions, integral transforms, convolution, and operational calculus will find here new results and their respective applications.
Many persons have made a significant contribution to this book, both directly and indirectly. Contribution of subject matter is duly acknowledged throughout the text and in the up-to-date bibliography.
We are especially thankful to Professors Hari M. Srivastava of the University of Victoria, Canada, Ivan H.Dimovski and Virginia S.Kiryakova of the Bulgarian Academy of Sciences and Shyam L.Kalla of the University of Zulia, Maracaibo, Venezuela for their keen support throughout the subject of this book, for suggest ing a number of invaluable improvements and for sending us relevant reprints and preprints of their works.
Finally we note, that this book was written during the academic year 1992 1993 at the "Research Scientific Laboratory of Applied Methods of Mathematical Analysis" of the Byelorussian State University, where both authors work.
October 1993 Byelorussian State University Minsk-220050, Republic of Belarus
xi
Chapter 1
Preliminaries
In this chapter we present notions and propositions concerning as special functions and integral transforms which will be necessary for further work.
1.1 Some special functions
Here we give definitions and simplest properties of some special symbols and func tions; more detailed information can be found in the books by A.Erdelyi et al. (1953), A.P.Prudnikov et al. (1989a) and F.W.J. Olver (1974).
A. Gamma-function f(z). The Euler integral of the second kind
(1.1)
is called the gamma-function. It is obviously convergent for all z E C for which ~(z) > 0. Here :l:z- 1 = e(z-l)ln(z). From the relation (1.1) we can derive the fact that the gamma-function is an analytic function in the half-plane ~(z) > °(see F.W.J. Olver (1974)). The gamma-function is extended to the half-plane ~(z) ~ 0, z -:f. 0, -1, -2, ... by analytic continuation of this integral. Namely, the reduction formula
f(z +1) = zf(z), ~(z» 0,
obtained from the relation (1.1) by integration by parts, yields the equality
f(z) = f(z+n) , z(z + l)(z +2) ... (z +n -1)
(1.2)
(1.3)
~(z»-n, n=I,2, ... , z-:f.O,-I, ... ,
which allows to carry out the analytic continuation into the half-plane ~(z) > -n for any n. The other method of analytic continuation is based on the Euler-Gauss formula
, z f(z) - lim n.n -J.
- n-tcoz(z+I) ... (z+n)' zrO,-I,-2, ... ,
1
(1.4)
2 Chapter 1
which can be obtained from the relation (1.1). The following useful estimate
Ir(z)1 :s jr(~(z))1 (1.5)
is a consequence of formula (1.4) (see F.W.J. Olver (1974». It follows from relation (1.3) that the function r(z) is analytic everywhere in the complex plane except z = 0, -1, -2, ... , where it has simple poles and is represented by the formula
(_I)1e r(z) = k!(z + k) [1 +O(z + k)], z -+ -k, k = 0,1,2,.... (1.6)
Here and everywhere in the book, the equality f(z) = O(g(z)), z -+ a means I~I < M < 00 as Iz - al < c. The relation f(z) = o(g(z)), z -+ a means that
lim &1.(z) = 0 and the equivalence f(z) ~ g(z), z -+ a means that lim &1.(z) = 1. From z-+a 9 Z .t--+a 9 :e
representation (1.6), we have
(_1)1e resz=_ler(z) =kl' k = 0,1,2, ....
We formulate some other properties of the gamma-function now: a) supplement formula
11" r(z)r(1 - z) = . ( );
(211") 2 Ie=O n
c) Weierstrass formula
1e=1
where I = lim (t ~ -In(n)) is the Euler's constant; n--+oo m=l
d) asymptotic Stirling formula
and its corollary
e) Dougall's formula
n=-oo
Preliminaries 3
11"2 [c+ d - a - b - 1 ] = sin(1I"a) sin(1I"b{ c - a, d - a, C - b, d - b '
~(a +b - c - d) < -1; a, b ¢ Z,
where, for convenience, we introduce the Slater's notation (L.J.Slater (1966), O.I.Marichev (1983))
(1.14)
In further discussion we will use the following formulae (see A.P.Prudnikov et al. (1986), Vol. 2. ) 1:00
r [a+u,b+u~c-u,d-U] du
[ a+b+c+d-3 ]
= rId 1 bIb d 1 ' ~(a +b+ c +d) > 3.a+c- ,a+ - , +c- , + -
1+00 [ -] 2c+d -
r du - =-:----=------:- -00 C +u, d - u - r(c + d - 1)
~(c +d) > 1.
r [a + iu~b - iU] du = Ta-br(a +b), 211" -00
~(a) > 0, R(b) > o. B. Pochhammer symbol (z)n with integer n is defined by the equality
r(z +n) n-l
(1.15)
(1.16)
(1.17)
(1.18)
From relation (1.18) and properties of gamma-function we obtain the following formulae
(l)n = n!,
(z)-n = ( )'l-z n
(zhn = 4n(z/2)n((1 + z)/2)n.
C. Binomial coefficients are defined for z, wEe by the formula
(z) nz + 1) = r( )r( )' z'l-I,-2, ....w w+l z-w+l
In particular, when z = n EN, w = mEN we have the equality
( n) n! m = m!(n - m)!"
(1.19)
(1.20)
(1.21)
(1.22)
(1.23)
(1.24)
4
Chapter 1
(1.25)
is called the beta-function. It is related to the gamma-function by the formula
B( ) _ r(s)r(t)
Using representation (1.25), we can obtain the following useful relations
B(s,t) =100
IB(s,t)/:::; B(!R(s),!R(t)), !R(s) > 0, !R(t) > O.
(1.26)
(1.27)
(1.28)
(1.30)
E. The generalized hypergeometric function pFq(z) is defined as a sum of the series
pFq[(a)p; (b)q; z] =pFq [«a))p;z] =pFq [a ll ,ap;z] (1.29) b q; bll ,bq ;
= :f: n~=1 (aj)n zn - n=O ni=l(bj )n n!'
The series on the right-hand side of relation (1.29) is absolutely convergent for all values of z, both real and complex, when p :::; q. Further, when p = q + 1, the series converges if Izl < 1. It converges when z = 1 if
and when Izl = 1, z f:. 1, if
For other values of z the generalized hypergeometric function is defined as an analytic continuation of this series. One of the methods of such a continuation is the Mellin Barnes integral representation
q+IFq [(a)q+I; (b)q; z]
_ ni=1 r(bj ) _1_11'+ioo ni~~ r(aj - s)r(s) (_ )-Od - +I . nq r(b ) z s,ni=1 r(aj ) 2n 1'-ioo j=1 j - s
where 0 < !R(s) = "Y < ~in !R(aj)j Iarg( -z)1 < 11". One can find complete list 1<)<q+l
of particular cases and p-;':operties of the generalized hypergeometric function in
Preliminaries 5
A.Erdelyi et al. (1953) and A.P.Prudnikov et al. (1989a). We will use the follow ing formulae in further discussion (see L.J.Slater (1966) and A.P.Prudnikov et al. (1989a)):
a) Gaussian summation theorem:
2F1 [a,b j 1] = r [c,c- a - bb] , !R(c - a - b) > 0. (1.31)
c; c - a,c-
Po [ -n,a,bj 1] _(c-a)n(c-b)n a 2 c,l+a+b-n-c - (c)n(c-a-b)n'
c) Dixon's summation theorem:
D [ a,b,c; ] a.£'2 1
(1.32)
(1.33)
= r [1 + a/2, 1 + a - b, 1 + a - c, 1 + a/2 - b - c] 1 + a, 1 + a/2 - b, 1 +a/2 - c, 1 + a - b - c '
!R(a - 2b - 2c) > -2.
F. Bessel functions Jv(z), Iv(z), Kv(z) are defined on the basis of the function
co n
oF1[c; z] = L -(z) ,= lim 1F1 [a; c; -aZ ] , Izi < 00, (1.34) n=O C nn. 0--+00
by the following formulae
1 (Z)V [ z2] Jv (z)=r(v+1) 2 oF1 v+1j-4
= ~ (_1)n(z/2)2k+v
co (/2)2k+v I v(z) = L z = e-wiv/2Jv(iz)
n=O rev +n + l)n!
(modified Bessel function),
sm 1I"V
Kn(z) = lim Kv(z), n = 0, ±1, ±2, ....v-+n (Macdonald function). It is clear that K_v(z) = Kv(z).
6 Chapter 1
Bessel functions Jv(:C), Iv(:C), Kv(:C) have the following asymptotic behaviour (see A.Erdelyi et al. (1953)):
Jv(:C) = (2cos(:c -11'(1 + Zv)/4) + O(:c-a/2 ), :c -+ +00,V;;
Jv(:C) = O(:cR(v»), :c -+ 0+,
Iv(:C) = O(:cR(v»), vi 0, :c -+ 0+,
Kv(:C) = O(e-Z jf;), :c -+ +00,
Kv(:C) = O(:c-1R(v)I), vi 0, Ko(:c) = O(ln(:c)), :c -+ 0+,
Here we also note the integral representation
(1.39)
(1.40)
(1.41 )
(1.42)
(1.43)
(1.44)
, where KiT(:c) is the Macdonald function with imaginary index, which we will need in further discussions (see A.Erdelyi et al. (1953)).
G. Mittag-Leffler function of vector index is an entire function defined by the series
(1.45)
=Lrr [ k)'Ie=O j=1 r OJ + f3j
where f3j > 0, ~(Oj) > 0, 1 ~j ~ n. Note that the Mittag-Leffler function of vector index is reduced to the Mit
tag-Leffler function EI/(z; 1') if n = 1 (for more details concerning Mittag-Leffler function we refer to A.Erdelyi et al. (1953)) and to the Bessel function of vector index if f3j = 1, 1 ~ j ~ n, which was investigated in details in M.I.Klyuchantzev (1983).
H. In an attempt to give a meaning to pFq(z) in the case p > q + 1 in 1941 C.S.Meijer introduced and studied the special function which is now well-known in the literature as the G-function and represented by the following Mellin-Barnes type of contour integral
am,n (zl(op)) = am,n (z/(oh,p) = ~ [ lII(s)z-'ds, p, q (f3q) p, q (f3h,q 211'l JL
where z i 0, 0 ~ m ~ q, 0 ~ n ~ p, OJ E C, 1 ~ j ~ p, f3j E C, 1 ~ j ~ q,
TIl=1 r(f3j + s) TI;=1 r(l - OJ - s) III(s) = TIP ( )TIq ( ) ,
j=n+! r OJ + s j=m+! r 1 - f3j - s
(1.46)
(1.47)
Preliminaries 7
an empty product, if it occurs, is taken to be one, and an infinite contour L separates all left poles 8 = -f3i - k, j = 1,2, ... , m, k = 0,1,2, ... of the numerator from the right ones 8 = 1 - CXi + k, j = 1,2, ... , n, k = 0,1,2, ... and under suitable conditions it may be one of the three types: L-oo , L+oo or L ioo (in particular, even a rectilinear line L = ('Y - ioo, 'Y +ioo». The description of contours and detailed list of properties and particular cases of the G-function may be found in A.P.Prudnikov et al. (1989a), Y.L.Luke (1969), A.M.Mathai and R.K.Saxena (1973).
We list here formulae of reflection and translation for the G-function:
(1.48)
(1.49)zQam,n (zl (CXp») = am, n (zl (cxp ) +cx) . p, q (f3q) p, q (f3q) + cx
I. In 1961 C.Fox introduced a more general function which is well-known in the literature as the Fox's H-function or the H-function. This function is also defined by the Mellin-Barnes type of contour integral as follows
(1.50)
(1.51)
where z f 0, 0 ~ m ~ q, 0 ~ n ~ p, CXi E C, ai > 0, 1 ~ j ~ p, f3i E C, bi > 0, 1 ~ j ~ q,
TIi:l r(f3i +bi 8) TIj=l r(1- cxi - ai8) '1)(8) = TIP ( )TI9 (f3 )'
i=n+l r CXi +ai8 i=m+l r 1 - i - ai8
an empty product, if it occurs, is taken to be one, and L is a contour in the complex s-plane, which is similar to the one in relation (1.46). If all ai, j = 1,2, ... ,p, and b;, j = 1,2, ... , q are equal to 1, then the kernel '1)(8) (1.51) is equal to \11(8) (1.47) and Fox's H-function (1.50) coincides with Meijer's G-function (1.46). Note, that the Mittag-Leffler function of vector index is a particular case of the H-function and it is reduced to the G-function if f3i, 1 ~ j ~ n are rational numbers. The H function was studied by various mathematicians and its properties are listed in the well-known memoir of B.L.J.Braaksma (1964), A.M.Mathai and R.K.Saxena (1978), and in the monograph by H.M.Srivastava et al. (1982). The formulae of reflection and translation for the H-function have the following form:
(1.52)
(1.53)
8 Chapter 1
1.2 Integral transforms
As it is known, the classical one-dimensional integral transforms are of the form
1 +00
[Kfl(:z:) = <p(:z:) = -00 K(:z:,t)f(t)dt, (1.54)
where K(:z:, t) is some given function (kernel of the transform), f(t) is an original in a certain space of functions, and <p(:z:) is the image of the function f(t). One of the most important integral transforms which will be the base of our investigations in this book, is the Mellin transform
roo j*(s) = M{f(t)js} = Jo f(tW-1dtj
its inverse is given by the formula
1 17 +
f(:z:) = M-1{f*(s)j:Z:} = -. j*(s):Z:-'ds, 'Y = R(s). 27l"t 7-ioo
(1.55)
(1.56)
The Mellin transform and its inverse are connected with the Fourier transform and its inverse by changes of variables and functions (seeO.I.Marichev (1983) ) and have wide applications. If we denote by --+ the correspondence between a function and its Mellin transform, then the following formulae of general type which we will use in further discussions can be easily proved
f(a:z:) --+ a-'j*(s), a> OJ
:z:Pf(:z:) --+ j*(s + p)j
f(:z:P) --+ 1~1j*(s/P)' P =J OJ
f (n)() r(n+1-s)f*( _ ) l' .-lc-1f(lc)( )-0:z: --+ r( ) s n, Im:z: :z: - ,1 - s .,...0
k = 0,1, ... , n - Ij
(d~:z:)n f(:z:)--+(I- stj*(s).
(1.57)
(1.58)
(1.59)
(1.60)
(1.61)
(1.62)
One can find the Mellin transform formulae of the majority of elementary and special functions in F.Oberhettinger (1974), O.I.Marichev (1983) and A.P.Prudnikov et al. (1989a)j here we list only Mellin transform formulae of some important func tions:
e-"P --+ 1~Ir(s/P)' R(s/p) > OJ (1.63)
Preliminaries
(1 _XP)~-1 r(s/p) r(a) -t Iplr(s/p + a)' ~(a) > 0, ~(s/p) > 0,
where (x)+ == H(x), H(x) is the Heaviside's function;
(xP _1)~-1 r(1 - a - s/p) 0 ~() 1 _ ~( / ). r(a) -t Iplr(1 _ s/p) , < a < s p ,
r(p)(1 + xtP -t r(s)r(p - s), 0 < ~(s) < ~(p);
1 r(s)r(1 - s) 0 IO() 1- -t ( / )' < :n. s < ,1r(I-x) r(s+I/2)r 1 2-s
sin(2y'i) r(s + 1/2) I~()I 1/2- .,fi -t r(1-s)' s < ,
r(s +1/2)r(-s) .,fierf(y'Z) -t r(1 _ s) ,-1/2 < ~(s) < 0;
r(s + v/2) Jv(2y'Z) -t r(1 + v/2 _ s)' -~(v/2) < ~(s) < 3/4;
2Kv(2y'Z) -t r(s + v/2)r(s - v/2), ~(s) > 1~(v)I/2;
-z/2K (~) r.:r(s+v)r(s-v) ~() I~( )1' e v 2 -ty1r r(s+I/2) , s> v,
eZ/2Kv(~) -t ~cos(1rv)r(s+v)r(s-v)r(I/2-S),
1~(v)1 < ~(s) < 1/2;
r(a) r(s)r(a - s) r(c) IF1(aj Cj -x) -t r(c _ s) ,0 < ~(s) < ~(a)j
E (_ ) -t r(s)r(1 - s) 11 X,JL r(JL-s/U) ,
o< ~(s) < 1, U> 1/2 or 0 < ~(s) < min{I,~(JL)/2}, U = 1/2;
r(a)r(b) F ( b' _ ) r(s)r(a - s)r(b - s) r(c) 21 a, ,c,-x -t r(c-s) ,
o< ~(s) < min{~(a), ~(b)}j
(1 - x)i--1 r(s)r(s + C - a - b) r(c) 2Fl(a,b;cjl-x)-t r(s+c-a)r(s+c-b)'
O<~(s), O<~(c), O<~(s+c-a-b);
111=1 r(aj) 111=1 r(aj - s)r(s) Il~ r(byFq[(ap)j (bq); -xl -t Il~ r(b. -) ,
3=1 3 3=1 3 S
o< ~(s) < 1~~ ~(aj), ble f 0,-1,-2, ,1:::; k:::; q _3_P
and
9
(1.64)
(1.65)
(1.66)
(1.67)
(1.68)
(1.69)
(1.70)
(1.71)
(1.72)
l)q = p - lor 2)q = p or
3) q = p + 1, ~(s) < 1/4 - 1/2 [~(~~=1aj - ~'=1 bj)] j
am,n (I(op)) ni=l f (f3j+s)l1i=l f (l-oj-s) (1.79) p, q Z (f3q) - n~=n+1 f(OJ + s) n'=m+1 f(1 - 13.1 - s)'
- ~in ~(f3j) < !R(s) < 1 - m!tx ~(Oj) 1::>3::>m 1::>3::>n
and 1) 2(m +n) > p +q or
2) (q - p)~(s) < q-~+1 +~ (~~=1 OJ - ~'=1 13.1)' 2(m +n) = p + q, Iq - pi 2: 2 or
3) m +n = p, q = p 2: 1, ~ (~~=I(Oj - 13.1)) > OJ
Hm,n ( I(op,ap)) ni=1 f(f3j + bjs) ni=1 f(1 - OJ - ajs) (1.80) p, q Z (f3q, bq) - n~=n+1 f(OJ + ajs) n'=m+1 f(1 - 13.1 - ajs)
- ~in ~(f3j)/bj < ~(s) < mjn (1 - ~(oj))/aj 1::>3::>m 1::>3::>n
and 1) t7 > 0 or
2) t7 = 0, c5~(s) < ? -1 +!R (~~=1 OJ - ~'=lf3j), where
n p m q q p
t7 = Laj - L aj +Lbj - L bj, c5 = Lbj - Laj. .1=1 j=n+l .1=1 j=m+l .1=1 .1=1
Comparing the right-hand sides of equalities (1.63)-(1.78) with the right-hand sides of (1.79), (1.80), it is not difficult to observe that the left-hand sides of the equalities (1.66)-(1.74), (1.76)-(1.78) ate particular cases of G-functions and the left-hand sides of the equalities (1.63)-(1.65), (1.75) are the particular cases of H-functions.
One can get a more detailed information about the theory of Mellin transform in classical spaces of functions in O.I.Marichev (1983) and E.C.Titchmarsh (1937). In particular, we will cite the following theorems, the proofs of whose can be found in E.C.Titchmarsh (1937).
Theorem 1.1 Let the function xe- 1 f( x) belongs to the space L(O, 00) of functions summable in the Lebesgue sense on the interval [0,00) and f( x) has the bounded variation in the neighbourhood of the point x = y. Let
1 +00
F(s) = M{f(t),s} = /*(s) = 0 f(t)t'-ldt, (s = e+ir). (1.81 )
Then HiT
f(x +0) + f(x - 0) = ~ lim 1 F(s)x-'ds. (1.82) 2 21l"z T-ooo (-iT
Preliminaries 11
(1.83)
Theorem 1.2 Let the function F(e + iT) belongs to the space L(-oo, +00) of functions summable in the Lebesgue sense on the interval (-00, +00) and has the bounded variation in the neighbourhood of the point T = t. Let
1 IHioo f(z) = M- 1{F(s)j z} = -2. F(s)z-'ds.
1U e-ioo
Then F(e + i(t + 0)) + F(e + i(t - 0)) - I' 1>' f( ) Hit- 1d- 1m y y y.
2 >'-+00 1/>' (1.84)
We will denote by Lp(p.(z ); R+), p ~ 1 the space of functions which are summable with power p on the interval (0, +00) and positive weight p.(z) in the Lebesgue sense; the norm is given by
{1 °O }l/P
IIfIlLI'(p(z);R+) = 0 p.(z) I f(z) IP dz < 00. (1.85)
In particular, when p.(z) == 1, we have the usual Lp-space. We note the known Holder inequality
where! + ! = 1 and the general Minkowski inequality p q
{1°O 1100 IP}l/P 100 {1°O }l/Po dz 0 f(z,y)dy ::; 0 dy 0 kf(z,y)IPdz .
(1.86)
(1.87)
We will recall also the Fubini theorem which allows to interchange the order of integration in iterated integrals.
Theorem 1.3 Let ill = [a, b], il2 = [e, d], -00 ::; a < b ::; 00, -00 ::; e < d::; 00, and let f(z, y) be a measurable function defined on ill X il2.
If at least one of the integrals
[ dz [ f(z,y)dy, [ dy [ f(z,y)dz, [ [ f(z,y)dzdy 101 102 102 101 101 102
is absolutely convergent then they coinside.
Theorem 1.4 Let f(z) E L2(z2e-1; R+). Then the function
/*(s,>.) =1>' f(y) yH it-1dy, (s=e+it) 1/>'
(1.88)
(1.89)
converges in the norm L2(e - ioo,e + ioo) to some function /*(s) and the function
1 I H i>.f(z, >.) = -2. /*(s)z-'ds
1U e-i>'
12
convf'.rges in the norm L2(x 2(-l;R+) to the function /(x), i.e.,
lim 100
>"_00 0
100
I f(y) 1 2 y2(-ldy = 2~ 1~ I /*(e + it) 1
2 dt.
Theorem 1.5 Let f(x) E L2(x 2(-l jR+), g(x) E L2(x l - 2(j R+). Parseval equality holds
1 00 1 l(+ioo
f(y)g(y)dy = -2. /*(s)g*(l - s)ds. o 1n (-ioo
Chapter 1
(1.90)
(1.91)
(1.92)
In further discussions we will also need the double Mellin transform. Let the function f(x,y) be defined on R~= (0,+00) x (0,+00). Then the classical double Mellin transform of the function f(x,y) at the point (s,t) E C 2 is the following integral
M{f(x,y)jS,t} = /*(s,t) =100100
f(x,y)x·- 1yt-1dxdy. (1.93)
This transform plays an important role in the theory of multiple integral transforms and convolutions. We give main properties of this transform.
Theorem 1.6 Let the function f(x,y) be continuous on R~ and the integral in the right side of (1.93) converges absolutely for all s, t, such that
101 < ~(s) < Ell 102 < ~(t) < E2 • (1.94)
Also, let the function f( x, y) be represented by the formula
f(x,y) = -(1')2 { {F(s,t)x-'y-tdsdt, (1.95) 21r~ JLc JLe
where Lc and Lc are vertical lines with real parts 0 and c, respectively (here 101 < c < E 1 ,f2 < 0 < E2). Then M{f(x,y)jS,t}= /*(s,t) = F(s,t) for all s,t such that !R(s) = c,~(t) = O.
Theorem 1.7 Let s, t belong to the strips defined by (1.94) and at these strips the following conditions hold
1. The function F(s,t) is analytic. 2. The following integral is convergent:
11 I F(s, t) I dsdt < +00 (1.96) LCi Lei
for all C1, 0 1 , such that
(1.97)
3. I F(s,t) 1-+ 0 for Is 1,1 t 1-+ +00. Then it follows from (1.95) that /*(s, t) = F(s, t) , where s, t belong to the strips
(1.94).
Preliminaries
The reader can find the proofs of these theorems in I.S.Reed (1944). The Mellin convolution
1+00 dt h(:;c) = (f *g)(:;c) = f(:;c/t)g(t)-
o t
13
(1.98)
(1.99)
will play an important role in our further considerations. It is well-known (see E.C.Titchmarsh (1937)), that if f(:;c):;ce- 1 E £(0,00) and g(:;c):;ce-1 E £(0,00), then h(:;c):;ce-1= (f *g) (:;c):;ce-1 E £(0,00) and h*(s) = /*(s)g*(s),s = e+iT. Using these facts and (1.56), we obtain the Parseval equality
1+00 dt 1 lHioo
f(:;c/t)g(t)- = -. /*(s)g*(s):;c-·ds. o t 21rt e-ioo
In this book, we will mainly consider integral transforms of the form
[Kf](:;c) = 100
K(:;c,t)f(t)dt. (1.100)
All classical integral transforms of the form (1.100) may be divided into two classes: Mellin convolution type transforms and transforms with respect to indices (or pa rameters) of special functions included in the kernels.
In the class of Mellin convolution type transforms, the following ones of the form (1.100) are best known: Laplace transform (k(:;c,t) = e-zt ), sine- and cosine Fourier transform (k(:;c, t) = y'2/1r sin(:;ct) and y'2/1r cos(:;ct)), Hankel transform (k(:;c, t) = VZtJIJ(:;ct)), Stieltjes transform (k(:;c, t) = r(p)(:;c+t)-P), Hilbert transform (singular integral) (k(:;c,t) = 1r-
1(t-:;ct1) in (1.54)), Meijer transform (k(:;c,t) = VZtKIJ(:;ct)), transform with Neumann function as kernel (k(:;c, t) = VZtY... (:;ct)), transform with Struve function as kernel (k(:;c,t) = VZtH... (:;ct)) , generalized Laplace transform with the function of parabolic cylinder as kernel (k(:;c, t) = 2.../2e-zt/2DIJ(~)), generalized Meijer transform (k(:;c,t) = (:;ct)I'-1/2e-zt/2W",I'(:;ct)), Buschman trans
form (k(:;c,t) = (:;c2 - t2r+A/2p;(t/:;c)), Riemann-Liouville integrals of fractional order: (left-hand sided (k(:;c,t) = (t - :;c)i--1/r(a)) and right-hand sided (k(:;c,t) = (:;c - t)i--1/r(a)), 1F1 -transform (k(:;c,t) = 1F1(a;c;-:;ct)), Gauss hyperge ometric transform (k(:;c,t) = r~(~~£b)2F1(a,b;c;-t/:;c)), Love transform (k(:;c,t)=
("'~(~r\F1(a, b; c; 1 - t/:;c)), Narain G-transform (k(:;c, t) =G;:;t (:;ctl ~~:n), et c. The class of transforms in respect to indices includes Kontorovich- Lebedev trans
form (k(:;c, t) = Ki",(t)), Mehler-Fock transform (k(:;c, t) = Pi~-1/2(t), t> 1; k(:;c, t) = 0, t < 1), Olevskii transform (k(:;c, t) = 2F1 (v-i:;c, v+i:;Cj 1+v-"l; -t)), Fa-transform (k(:;c, t) = (t _1)~-2"'-1J Fa(1- v-,8 -i:;c, a', 1- v -,8 +i:;c,b';2 - 2v -,8; I-t, 1- t)), Lebedev transforms (k(:;c,t) = [Ii",(t) + Li",(t)]Ki",(t), k(:;c,t) = Ki~(t)), Wimp
transform (k(:;c,t) = a;+~~2 (W-IJ+;""(i~-i""(QP»)) , Lebedev-Skalskaya transforms
(k(:;c,t) = {:} K1/2+i",(t)), et c. The abovementioned transforms and their applications have been considered in
great variety of monographs, books, and papers and the difficulty of studying the
14 Chapter 1
theory of integral transforms resides in the fact that each transform needs a specific methods for investigations. In this book, we will try to present the general approach to the theory of integral transforms.
Chapter 2
2.1 General Fourier kernels
In this section, following G.N.Watson (1933), E.C.Titchmarsh (1937), Vu Kim Tuan (1986a), Vu Kim Tuan (1986b), Vu Kim Tuan (1987), Vu Kim Tuan et al. (1986), we consider the Mellin convolution type transforms of the form
g(x) = [Kf](x) =100
f(x) = [Kg](x) =100
k(xy)g(y)dy,
(2.1)
(2.2)
(2.3)
where the kernel k(x) is called the conjugate kernel. We will investigate these transforms in the special functional space M-1(L)
introduced by Vu Kim Tuan et al. (1986). As it is showed below, this space is very convenient for studies of transform (2.1).
Definition 2.1 Denote by M-1(L) the space of functions f(x), x > 0, repre sentable by inverse Mellin transform of integrable functions r (s) E L1(q) == L(q) on the contour q = {s E C : ~(s) = 1/2}:
f(x) = M-1{/*(s); x} = -2 1 .1 /*(s)x- 6 ds. 1n (T
The space M-1(L) with the usual operations of addition and multiplication by scalar is a linear vector space. If the norm in M-1(L) is introduced by the formula
1 1+00
15
(2.4)
16 Chapter 2
then the space M-I(L) is a Banach one. Now we consider the main properties of the space M-I(L). 1) f(x) E M-I(L) if and only if x-If(x-I) E M-I(L) . This property is a result of the fact that the functions f( x) and X-I f( X-I) are
the inverse Mellin transforms of the functions /* (s) and /*(1- s) respectively, which simultaneously either do or do not belong to L(O").
2) If f(x) E M-I(L) then XI/2 f(x) is uniformly bounded, continuous on (0, +(0), and, furthermore, XI/2f(x) = 0(1), when x -+ +00 and x -+ O.
This property follows from the Riemann-Lebesgue lemma (see E.C.Titchmarsh (1937) ).
3) If f(x), g(x) E M-I(L), then xl / 2 f(x)g(x) E M-I(L). This follows from the fact, that X
I /
2 f( x )g(x) is the inverse Mellin transform of the function 2;. ftT /*(r)g*(s -r+ 1/2)dr, which belongs to L(O") by Fubini theorem. Here, g(x) = M-I{g*(s)jx}.
4) Let f(x) E M-I(L) and X- I/2g(X) E L(R+). Then fooo g(U)f(;)d; belongs to M-I(L).
In fact, by the property of Mellin convolution, this integral is an inverse Mellin transform of the function /*(s)g*(s) and since /*(s) E L(O") and g*(s) belong to the space of essentially bounded functions on 0" (Loo(O")), then /*(s)g*(s) E L(O").
5) Let f(x) be twice differentiable function and moreover x3/2f"(x) E L(R+) and XI/2 f( x), X3
/ 21'(x) tend to zero, when x tends to zero and infinity. Then
f(x) E M-I(L). The proof of property 5) follows from the estimate of the Mellin transform (1.81)
of the function f(x) by using the integration by parts and Theorem 1.1.
Definition 2.2 Denote by J(, the set of kernels k(x) for which the following condi tions hold
1) k(x) E L(f,E) for any f, E, such that 0 < f < E < 00 ;
2) The integral
k(u)u'-Idu, s E 0" (2.5)
is convergent and there exists the constant G > 0, such that for almost all f, E > 0 and t E R the following estimate is satisfied
(2.6)
If for the kernel k(x) E J(, there exists the conjugate kernel k(x) E J(" such that the equality
k*(s)k*(1 - s) = 1 (2.7)
holds almost everywhere on the line ~(s) = 1/2, then we will say that k(x),k(x) E J(,* C J(,.
It is evident, that if X- I/2k(x) E L(O, (0), then k( x) E J(, and k(x) f/ J(,*.
Transforms With Arbitrary Kernels 17
Theorem 2.1 (Vu Kim Tuan (1986b» Let f(z) E M-1(L), k(z) E Je. Then the following Parseval formula takes place
[00 k(zy)f(y)dy = -2 1 .1 k*(s)/*(1 - s)z-·ds.
Jo ~~ u
Proof. Using relation (2.3), we have the chain of equalities:
(2.8)
(2.10)
[00 k(zy)f(y)dy = lim -2 1. [N k(zy) 1 /*(1 _ s)y·-1dsdy
Jo M,N.....oo ~~ J1/M u
M.N.....oo ~~ u "'1M
= -2 1 .1 k*(s)/*(1 - s)z-·ds. ~~ u
We obtain the last equality by means of Lebesgue theorem, which is applicable in this situation, since the inner integral in the left part of the equality converges to the function k*(s) because of relation (2.5) and this function is bounded almost everywhere as a result of the estimate (2.6), therefore the function k*(s)f*(I- s) E L(CT).•
Now we consider the integral transform (2.1) in which the kernel k(z) E Je*.
Theorem 2.2 (Vu Kim Tuan (1986b» Let k(z) E IC*, and let k(z) E Je* be its conjugate kernel. Then integral transform (2.1) is an automorphism in the space M-1(L) and its inverse has the form (2.2).
Proof. Using the Parseval equality (2.8), we may rewrite the relation (2.1) in the form
g(z) =~1k*(s)/*(1 - s)z-·ds. (2.9) 2~~ u
"- Since the function k*(s)f*(1 - s) E L(CT) (see the proof of Theorem 2.1), then g(z) is an inverse Mellin transform of the function g*(s) = k*(s)f*(1 - s) E L(CT) and therefore g(z) E M-1(L). Furthermore, we have
IIg(z)IIM-l(L) ~ II k*(s)IILoo(u)lIf(z)IIM-l(L),
IIk*(s)IILoo(u) < +00. The function k*(1 - s) belongs to the space of essentially bounded functions on CT, and the functions k*(s) and k*(1 - s) satisfy relation (2.7). Consequently, the function l/k*(s) is bounded on CT, the function f*(s) = g*(1 - s)/k*(1 - s) E L(CT), and f(z) E M-1(L). Using the Parseval equality (2.8), we obtain
18
Chapter 2
(2.12)
Corollary 2.1 Let k(x) E Je*, let k(x) E Je* be its conjugate kernel, and Ik*(8) 1= 1,8 E a. Then the integral transforms (2.1) and
(2.2) are i80metric automorphisms in the space M-1(L).
To prove Corollary 2.1, it is sufficient to use Theorem 2.2 and the following equalities:
I/g(X)I/M-l(L) = I/g*(8)I/L(CT) = I/k*(8)1*(1 - 8)I/L(CT) (2.13)
= 1I1*(8)IIL(CT) = IIf(x)IIM-l(L)'
2.2 Examples of the Fourier kernels
In this section, using the results of the previous section, we consider some examples of general Fourier kernels, both known and new ones. Example 2.1 Let us consider the sine-Fourier transform
[F.fl(x) = ../2/7r100
sin(xy)f(y)dy. (2.14)
We will show that the kernel k(x) = ../2/7rsin(x) of this transform belongs to the set Je*, I k*(8) 1= 1 and the conjugate kernel k(x) E Je* coincides with k(z). In fact, using relation (1.68), we obtain
k*(8) = ../2/7r100
y./2-1 sin(2VY)dy (2.15)
_ ._1/2 f (8/2+1/2) - 2 r(1 _ 8/2) ,8 E a.
Furthermore, we have
+ liE sin(u)uit-l/2dul =1 I 1(t) I + I12(t) I·
To estimate I 1(t) and 12(t), we note that the integral
11 sin(u)uit- 1/2du
is absolutely convergent and the integral
100
sin(u)uit -
1 /
2du
is uniformly convergent according to the Abel test of uniform convergence of integrals and, consequently, there exists the absolute constant A, such that for any B 2:: A and for all t E R we have
(2.17)
(2.22)
U- 1 /
2du = 2. (2.18)
Using the estimate (2.17), we have the following estimates for I12(t) I: 1) if E :::; A, then
I12(t) I:::; 11E
I12(t) I:::; 11A
Comparing estimates (2.16) and (2.18)-(2.20), we obtain that k(:z:) = y'2/1r sin(:z:) E JC. It follows from relation (2.15) that
k*(s)k*(I- s) = 2_-1/2f(S/2 + 1/2)21/2-_ f(l- s/2) = 1. (2.21) f(1 - s/2) f(s/2 +1/2)
Therefore, k(:z:) E JC* and the conjugate kernel k(:z:) E JC* coincides with k(:z:). We have also
1 k*( ) 12= k*( )k*( ) = 2itf(it/2 + 3/4)
s s s r(3/4 _ it/2)
T itf(-it/2+3/4) -1 X f(3/4 + it/2) - .
Usingrelations (2.21), (2.22) and Corollary 2.1, we obtain finally, that the sine Fourier transform (2.14) is an isometric automorphism in the space M-1(L) and its inverse has the same form. Example 2.2 Let us consider the modified Hankel transform
(Jvf)(:z:) =100
20 Chapter 2
Using equality (1.70), we obtain the following relation for the kernel k(:c) = J,,(2y'Z)
The estimate
*() 100
( r.:) 0-1 f(s+v/2) k s = 0 J,,2 y uu dU=f(1+v/2_s)' SECT. (2.24)
liE J,,(2.jU)uit-1/2dul :::; C (2.25)
may be proved by using the same technique as in Example 2.1, and the asymptotic behaviour of the Bessel function in the neighbourhood of the points 0 and 00 (see formulae (1.38) and (1.39)). We have also
k*(s)k*(1- s) = f(s +v/2) f(1 - s + v/2) = 1. (2.26) f(1 + v/2 - s) f(v/2 + s)
Therefore, k(:c) E lC* and the conjugate kernel k(:c) E lC* coincides with k(:c). If v E R, then the following equality takes place:
I k*(s) 12= k*(s)k*(s) = f(it + 1/2 +v/2) (2.27) f(1/2+v/2 -it)
f( -it + 1/2 + v/2) X =1.
f(1/2 + 11/2 + it) Using relations (2.26), (2.27), Theorem 2.2, and Corollary 2.1, we find that the modified Hankel transform (2.23) is an automorphism in the space M-1(L), its inverse has the same form, and, furthermore, if v E R, then the modified Hankel transform (2.23) is an isometric automorphism in the space M- 1 (L).
We proved that J2/,Tfsin(:c), J,,(2y'Z) belong to the class lC* of kernels. These statements are particular cases of the following general
Theorem 2.3 (Vu Kim Tuan (1987» Let
q - m - n = m + n - P = 11/2 =I 0, (2.28)
~ (~aj - t,Bj) = 0, (2.29)
~(aj) < 1/2, 1:::; j :::; nj ~(,Bj) > -1/2, 1:::; j :::; mj (2.30)
~(aj) > -1/2, n + 1 :::; j :::; Pj ~(,Bj) < 1/2, m + 1 :::; j :::; q. (2.31)
Then the function
k(:c) = a;:~n (:cl ~;:D (2.32)
belongs to the class lC* of kernels and its conjugate kernel has the form
k(:c) = Gq-m,p-n (:c1-(a:+1)' -(ai ») (2.33) p,q - (,B~+l)' - (,Bi" ) ,
where the symbol (af) denotes the set (ai, ai+l, ai+2,' .. ,aj).
Proof. First we consider the case "I > O. According to O.I.Marichev (1983), the G-function (2.32) has the following asymptotic behaviour under conditions (2.28) (2.31)
(2.34) {
O(XC Iln(x) 1m - I ), X -+ 0+, c = min 'iR(f3j) > -1/2;
k(x) = l~J~m
O(xa - l Iln(x) In-I) + AxPcos("IXl/'1 + <5), X -+ +00,
where a = max 'iR(Oj) < 1/2, p = (1- "1)/(2"1), A and <5 are some constants. Then l~J~n
we have
=1 Il(t) I+ I12(t) I;
I Il(t) 1= IiI O(UCIln(u) Im-l)uit-l/2dul (2.36)
S; 11
+ liE AuPcOS("IUI /'1 + <5)uit-I/2dul =1 12l(t) I+ I122(t) I .
Choosing any t, such that 0 < t < 1/2 - a, we obtain
(2.40)
since the integral floo cos("Iu+t5)ui'1t-I/2du is uniformly convergent according to Abel test of uniform convergence of integrals.
Comparing the estimates (2.35)-(2.39), we obtain
liE k(u)uit-I/2dul ::; C
for 0 < (; < E < 00 and t E R. The same estimate can be obtained for the function k(x), as well.
Using relation (1.79), we have
k*( ) = 0;:1 f(f3j + s) OJ=1 r(1 - (Xj - s) s E u, (2.41) s O~=n+I r(OJ + s) OJ=m+I r(1 - f3j - s)'
22 Chapter 2
(2.42) k*( ) _ IU=m+I r(s - f3j) II~=n+l r(1 + OJ - s)
s - IIi=1 r(s _ OJ) II;:l r(1 + f3j _ s) , s E u.
It follows from relations (2.~0)-(2.42), that k(:c), k(:c) E JC and equality (2.7) is true, that is, k(:c) E JC* and k(:c) is its conjugate kernel.
In the case 1J < 0, we use formulae (1.48), (1.49) of reflection and translation for G-function and obtain
j E Gm,n (ul(op)) Uit-l/2du = 11 /. Gn,m (yI1- (f3q)) y-it-3/2d
p,q (f3 ) q,p 1 - (0 ) y • q l/E P
(2.43)
(2.44)
= r/' G;,-: (yl-(f3q)) y-it-l/2dy. ll/E -Cop)
Therefore, we reduce the case 1J < 0 to the case 1J > 0, for which the statement of theorem has been proved.•
Corollary 2.2 Let conditions (2.28)-(2.31) hold and f3 > O. Then the function
k(:c) =f3:c~/2-1/2G;,~n (:c~I~;:D
belongs to the class JC* of kernels and its conjugate kernel has the form
(2.45)
The statement of this corollary follows from the fact, that f3:c~/2-l/2k(:c~) E JC* if k(:c) E JC* , which can be verified without any difficulties.
Remark 2.1 The contour u = {s E C : ~(s) = 1/2} separates the left poles of k*(s) and k*(s) (see formulae (2.41), (2.42)) from the right ones due to the conditions (2.30), (2.31). This separation is required by definition ofG-function. We can make the conditions (2.30), (2.31) weaker and rewrite them in the form
!R(Oj):rH/2+k, l~j~n, k=O,l, ... j (2.46)
(2.47)
~(oj)f=-1/2-k, n+l~j~p, k=O,l, j
~(f3j) f= 1/2 + k, m + 1 ~ j ~ q, k = 0,1, ..
In this case the functions k(:c), k(:c), which are determined by the inverse Mellin transform of the functions (2.41),
(2.42), differ from the G-functions in the right parts of formulae (2.32), (2.33) by a finite power series but the statement of the Theorem 2.9 remains true.
Transforms With Arbitrary Kernels
Theorem 2.4 (Vu Kim Tuan (1987» Let
m i' n; aj E R, aj < 1/2, 1 ~ j ~ n;
f3j E R, f3j> -1/2, 1 ~ j ~ m.
Then the integral transform
() roo ~,n ( I(an), -(an)) f( )d 9:C = Jo 2n,2m :Cy (13m), -(13m) y y
23
(2.48)
(2.49)
is an isometric automorphism in the space M- 1(L) and its inverse has the same form.
Proof. Using condition (2.48), we can apply the Theorem 2.3 and obtain, that the
kernel k(:c) = G~~m (:cll;.:~: =~~:n of the transform (2.49) belongs to the set /C*
and its conjugate has the same form. Furthermore, we have
- II'"!' r({3· + s) II~ r(1 - a' - s) 1k*(s) 12= k*(s)k*(s) = ~=1 3 ~1 3 (2.50)IIj=1 r(s - aj) IIj=1 r(1 +{3j - s)
IIi=1 r(s - aj) IIi=1 r(1 +{3j - s) x IIm ( ) IIn ( ) = 1, s E U.j=1 r {3j + s j=1 r 1 - aj - s
Using relation (2.50) and Corollary 2.1, we obtain the statement of the theorem.• We will consider now Parseval equality for the integral transforms (2.1), (2.2).
Theorem 2.5 (Vu Kim Tuan (1987» Let k(:c) E /C* and let k(:c) be its conju gate kernel, f(:c) E M- 1(L), and :c-1/ 2f(:c) E L(O, 00),
91e(:C) =100
Proof. First we consider the following integral
100 lEY= f(y)y-' k(u)u·- l dudy. o ey
(2.51)
(2.52)
(2.53)
(2.54)
(2.55)
It follows from relation (2.54) and k(:z:) E JC, :z:-1/2f(:z:) E L(O, 00) that glc(:Z:) E JC and g*(s) = k*(s)/*(1 - s). Using Parseval formula (2.8) and Lebesgue theorem, we have
l CX> 1 iN 1 Aglc(u)g;.(u)du = lim -2' glc(u) k*(s)
o M,N-.cx> 1n 11M "
x/*(l - s)u-·dsdu = lim ~1k*(s)/*(l - s) iN glc(u)u-·duds M,N-.cx> 211"t " 11M
= -2 1 ·1 k*(s)/*(l - s)k*(l- s)/*(s)ds = ~ 1 /*(1 - s)/*(s)ds
1I"t " 211"t "
= -2 1 .1/*(1- s) lCX> f(u)u·- l duds = lCX> f(u)
1I"t ,,0 0
x- 2 1
.1/*(1 - s)u·- l dsdu = lCX> P(u)du. 1I"t " 0
The Theorem 2.5 is proved.• Now we consider some examples of transforms having the form (2.1), (2.2).
Example 2.3 The integral transform
l CX> (sin(:z:y) )
g(:z:) = 0 :z:y +cos(:z:y) f(y)dy
is an automorphism in the space M-l (L) and its inverse has the form
21CX>f(:z:) = -- :z:ySi(:z:y)g(y)dy, 11" 0
where Si(:z:) is the integral sine (see A.P.Prudnikov et al. (1986)). We have the following representation of the kernel of transform (2.56):
sin(:z:) (1I":Z:) 1/2 k(:z:) = -:z:- +cos(:z:) = 2
(2.56)
(2.57)
(2.58)
11 (:z:21 -5/4 ) XGl :3 4" -1/4, -1/4, -3/4 .
Using Theorem 2.2, Corollary 2.2, and Remark 2.1, we find that the integral trans form (2.56) is an automorphism in the space M-l (L) and its inverse has the kernel of the form
k(:z:)= .fi l r [s+1/4,s+3/4] (~)-2·ds V211"3/2i" s + 5/4,3/4 - s 2
= _ 2:Z: Si (:Z:). 11"
(2.59)
(2.60)
where Ci(xJis the integral cosine (see A.P.Prudnikov et al. (1986)) is an automor phism in the space M-1(L) and its inverse has the form
100 (1 - cos(:z:y) . ) f(:z:) = 0 :z:y + sm(:z:y) g(y)dy. (2.61)
We have the following representation of the kernel of transform (2.60):
2:z:. (2:Z:) 1/2 11 (:z:21 5/4 ) k(:z:) = -;-Cl(:Z:) = -;- G1:3 4 1/4, 1/4, 3/4 . (2.62)
Using Theorem 2.2, Corollary 2.2, and Remark 2.1 we obtain that the integral transform (2.61) is an automorphism in the space M-1(L) and its inverse has the kernel of the form
(2.63)
(2.64)
Finally, let us consider transform (2.1) from the point of view of general Fourier transforms (see E.C.Titchmarsh (1937)) in the space L 2(R+) == L 2(0, 00)
g(:z:) = [Kf](:z:) = l.i.m'N-+oo iN k(:z:y)f(y)dy,
where convergence of the integral (2.64) is understood in the L2-sense. In this case, we have the inverse transform in the form
f(:z:) = [Kg](:z:) = l.i.m.N-+oo iN k(:z:y)f(y)dy, (2.65)
(2.66)[Fcf](:z:) = J2/7r l.i.m.N-+oo iN cos(:z:y)f(y)dy,
and the Hankel transform
where for the conjugate kernel k(:z:) and the kernel k(:z:), condition (2.7) is valid. Such kernels k(:z:), k(:z:) E K;* are called general Fourier kernels. The well-known examples of transforms (2.64), (2.65) are the sine-Fourier transform (2.14), the cosine-Fourier transform
for which the inverses have the same forms. The operators (2.64), (2.65) are bounded in the space L2 (R+) and in order to obtain the corresponding Plancherel theorems we introduce a notion of Watson kernels.
26 Chapter 2
A function kl (:z:) is called a Watson kernel if
:Z:. 11 /2+
1n 1/2-iN 1 - s
where the function k*(s) is defined by relation (2.5). Examples of Watson kernels are the functions J2(rrsin(:z:), J2!ir(1- cos(:z:», 1/11' In I~I, l/ll'ln 11- z 2 1. If k l (:z:) is a differentiable function on half-axis (0, 00), then it is not difficult to see that k~(:z:) = k(:z:), where k(:z:) is a Fourier kernel. A function kl (:z:) is called a conjugate Watson kernel if
A :Z:. 11/2+iN k*(s) _. kl (:z:) = -.l.l.m.N-+oo --:z: ds
2n 1/2-iN 1 - s (2.69)
(2.70)
(2.71)
(2.73)
(2.72)
and equality (2.7) takes place. Since the functions k*(s), k*(s) are bounded on the line !R(s) = 1/2, then the integrals (2.68), (2.69) exist in the L 2-sense and in accordance with Theorem 1.4, kl (:z:)/:z:, kl (:z:)/:z: belong to L 2(R+).
Hence, by the Mellin-Parseval equality of type (1.99) we have
1 11
~ /*(s)g*(l - s):Z:-'ds = f(:z:y)g(y)dy, 1l'~ 1/2-ioo 0
where f(:z:), g(:z:) E L2(R+) and /*(s), g*(s) are their Mellin transforms in L2(R+). We can evaluate the next improper integral (so-called Watson condition)
100 kl(:z:u)kl(YU)d . ( ) 2 U = Illln :z:, y ,
o u
which provides the boundedness of the corresponding Watson transforms pair in L 2 (R+)
d 100
A d 100 A dyf(:z:) = [Kg](:z:) = -d kl (:z:y)g(y)-,
:z: 0 y
where the integrals (2.72), (2.73) are understood as improper.
Theorem 2.6 Let k(:z:), k(:z:) E JC* and kl (:z:), kl (:z:) be the corresponding Watson kernels (2.68), (2.69). Then for any function f(:z:) E L2(R+), its general Fourier transforms (2.64), (2.65) [Kf](:z:), [Kf](z) belong to L2(R+) and almost everywhere on R+ the dual equalities (2.72), (2.73) are valid. Moreover, the Parseval equality
l°O[Kf](u)[Kf](u)du = 100
f2(y)dy
100
1 f(y) 1 2
I f(y) 1 2 dy,
where c, c are some constants.
(2.74)
(2.75)
(2.76)
Proof. Let f(x) E L2(R+) and f*(s), s E u be its Mellin transform (1.81) in L2(R+) such that in accordance with Theorem 1.4 f*(1/2 + it) E L2(R). Since k(x), k(x) E JC*, then k*(1/2+it), k*(1/2+it) are bounded functions and we have that the products k*(1/2+ it)f*(1/2 - it), k*(1/2 + it)f*(1/2 - it) belong to L2(R). Hence, from the Mellin-Parseval equality (2.70) it follows that [Kf](x), [Kf](x) are their inverse Mellin transforms, which are from L 2(R+) and by the Parseval equality, it is not difficult to obtain the inequalities (2.75), (2.76) and the representations
1 11 /2+
[Kf](x) = -2.1.i.m.N_00 k*(s)f*(I- s)x-·ds, ~t 1/2-iN
(2.77)
(2.78) , 1 11/2+iN,
[Kf](:c) = -.l.i.m.N_oo k*(s)f*(1 - s):c-·ds. 2~t t/2-iN
Further, using Theorem 1.5 for [Kf](y) and g(y) = 1, 0 < y ~ x, g(y) = 0, y> x and evaluating g*(I- s) = x t -·/(I- s),s E u in equality (1.92), we have
l "'[Kf](y)dy = 2 X .1 k
1 *(s) J*(I- s):c-·ds.
o ~t 0' - S (2.79)
From the Mellin-Parseval equality (2.70), the right side of relation (2.79) can be represented as follows
x 1k*(s) 100
dy-2' -1-J*(1 - s)x-·ds = kt(:cy)f(y)-· ~t 0' -s 0 Y
(2.80)
Hence, almost everywhere on R+, we have relation (2.72) and analogously the for mula (2.73). Therefore, the general Fourier transform (2.78) of the function g(:c) defined by formula (2.72) is the inverse Mellin transform of the following function
k*(s)k*(1 - s)J*(s) = J*(s) (2.81)
in accordance with the condition (2.7). Thus we obtained the dual formulae (2.72), (2.73).
Finally, in order to establish the Parseval relation (2.74), we use Theorem 1.5. From the chain of equalities
roo [Kf](u) [Kf] (u)du = -2 1 .1 k*(s)k*(1 - s)J*(s)J*(I- s)dsJo ~t 0'
= ~1J*(s)J*(1 - s)ds = 100
f2(y)dy 2~t 0' 0
it follows that relation (2.74) is valid. The Theorem 2.6 is proved.•
Corollary 2.3 The Watson kernels k1(x)/x, kt(x)/x belong to L2(R+) and condi tion (2.71) holds.
28 Chapter 2
Proof. Since, according to equalities (2.68), (2.69), the functions k1(a;)/a;, k1(a;)/a;
are the inverse Mellin transforms of the functions k;i:>, ~'i:>, s = 1/2 + it, t E R,
which belong to L2(R), the Watson kernels k1(a;)/a;,k1(a;)/a; are from L2(R+), and by the Cauchy residue theorem, we have
tX> k1(a;u) k1(yu) du = -1-1 k*(s)a;-· k*(l - S)y·-l ds (2.82) 10 a;u yu 211"i iT 1 - s s
= -1-1 a;-.y.-l ds = {1/a;, y ~ a;, = min(a;, y). 211"i iT (1 - s)s l/y, a; < y a;y
Thus we get condition (2.71).•
Corollary 2.4 If k(a;) = k(a;), then we have the following Parseval equality for the Mellin convolution type transform (2.1)
(2.83)
Theorem 2.7 Relation (2.71) is fulfilled for all a;, y E R+ iff there ea;ist the dual formulae (2.72), (2.73) in L2(R+).
Proof. First we assume that f (a;) belongs to the space of Coo functions with compact support in (0,00). Then the functions [K fl(a;), [K f]( a;) are differentiable functions and it is not difficult to obtain the following representations
Thus we have,
, 11 00
1 00
[Kf](a;)[Kf](a;)da; = 2da; k1(a;y)f'(y)dy k1(a;u)f'(u)du o 0 a; 0 0
1 00 100 100 , da; 100
= j'(y)dy j'(u)du k1(a;y)k1(a;u)2 = j'(y)dy 000 a; 0
x100 f'(u)min(y,u)du = 100
f'(y)ydy100 f'(u)du +100
x100 j'(y)dy = -2100
f(u)j'(u)udu = 100 f2(u)du.
Now let f(a;) be an arbitrary function from L2(R+). As it is known, the space of Coo functions with compact support is dense everywhere in L2(R+). Hence there exists a sequence of functions fn( a;) such that
lim [oo[f(u) - fn(uWdu = o. n-too 10
By means of the Parseval equality (2.74), we have the relation
29
Therefore
(2.86)
both sides of which tend to zero when m, n - 00 since {fn} is a Cauchy sequence. It follows from inequalities (2.75), (2.76) that the sequences {[K fn]}, {[K fn]} are square convergent and we obtain
Hence we have
1z 1z 100
•[Kf](u)du = lim [Kfn](u)du = lim k1(xy)fn(Y)- o n-oo 0 n_oo 0 Y
100 dy = k1(xy)f(y)-
and obtained the representation (2.72). Let
[Kt/J](x) = ~1"" k1(y)dy = k1 (:r:u)
dx 0 y x and by applying the Parseval equality (2.74), it is not difficult to obtain the dual formula (2.73).
Finally, we will prove that conditions (2.7) and (2.71) are necessary ones. Let us suppose that the dual formulae (2.72), (2.73) are valid for any function
f(x) from L2(R+). We take f(u) = 1, u :S x, f(u) = 0, u > x. Then [Kf](u) = k1(xu)/u and analogously [Kf](u) = k1(yu)/u. Substituting these transforms into relation (2.74), it is not difficult to obtain relation (2.71).
Moreover, if we take y = 1 in relation (2.82), we can obtain the following identity for all x> 0
_1_1_1_-_k-:-,*(--"-s)_k*..".:..(1_-~s)x-Ids = O. 211'i CT (l-s)s
Since the integrand expression in (2.86) belongs to L(u) as a product of two functions from L 2(u), it follows from the inversion Mellin transform formula that the integrand is equal to zero, Le., we have condition (2.7). Theorem 2.7 is proved.•
30 Chapter 2
2.4 Bilateral Watson transforms
In this section, we will generalize the Watson transforms (2.72)-(2.73) introduced in unitary case (see the Corollary 2.4), when k(:z:) = k(:z:), onto real axis R (Vu Kim Tuan and S.B.Yakubovich (1992». From Section 2.3 it follows that in this sym metrical case the Watson transform (2.72) is unitary and has the inversion formula (2.73), where k(:z:) = k(:z:) if and only if the condition (2.71) is fulfilled.
Concerning the bilateral Watson transforms (the case of the real axis (-00, +00», the classical examples of such transforms are the Fourier transform in L2(R)
d 1+00 ei",y - 1
g(:z:) = d f(y)dy, :z: -00 y
and the Hartley transform pair (see R.N.Bracewell (1986»
r+M
r+M f(:z:) = (21rt 1
/ 21.i.m'M,N-+00 l-N (cos(:z:y) + sin(:z:y»g(y)dy.
(2.87)
(2.88)
(2.89)
Using the properties of Watson kernels, we give the criterion of unitarity of the general bilateral Watson transform
d 1+00
dy(Kf)(:z:) = g(:z:) = d k(:z:y)f(y)-, :z: -00 y
which can be applied to obtain the new Watson transform pairs.
(2.90)
Theorem 2.8 The bilateral integral transform (2.90) is unitary in the space L2 (R) and the symmetrical inversion formula
- d 1+00 -- dy(Kg)(:z:) = f(:z:) = -d k(:z:y)g(y)-
:z: -00 y
holds if and only if the kernel k(x) possesses the following representation
(2.91)
(2.92)
where k~i)(:z:), :z: E R+, i = 1,2 are the Watson kernels (2.68), moreover k~i)(:z:) = k~i) (:z:) and relation (2.71) holds.
Proof. From equality (2.92), we have
k(:z:) = T 1 ([k(:z:) + k(-:z:)] + [k(:z:) - k(-:z:)]) = (2.93)
= T 1 ([k(1 :z: /) +k( - I:z: I)] +sgn(:z:)[k(l :z: /) - k( - I:z: I)]).
We introduce the quantities
k~l)(1 Z I) = k(1 z I) + k(-I z I), k~2)(1 z I) = k(1 z I) - k(-I z I). (2.94)
We will establish that the functions k~i)(1 z I), i = 1,2 are Watson kernels (2.68). To do this, using formula (2.92) we represent the function j(z) as follows
(2.95)
where
j~l)(1 z I) = j(1 z I) - j(- I z I), j~2)(1 z I) = j(1 z I) + j(- I z I). (2.96)
Taking the transform (2.90)
d /+00 dy(Kf)(z) = {k(z); j(z)} (z) = -d k(zy)j(y)- z -00 y
(2.97)
= {T1[kp)(1 z I) + sgn(z)k~2)(1 z I)]; Tl[sgn(z)j~l)(1 z I) + j~2)(1 z I)]} (z)
= 2-2 {kP)(1 z I); sgn(z)j~l)(1 z I)} (z) + T 2 {sgn(z)k~2)(1 z I); j~2)(1 z I)} (z).
By evenness, we have
{sgn(z)k~2){1 z l);sgn(z)f~l)(1 z I)} (z)
d /+00 dy= -d sgn(zy)k~2){1zy I)sgn{y)j~l)(1 y 1)- = O. z -00 Y
Now, if [K(l)j]{Z) and [K(2)j]{Z) are integral transforms (2.72) with kernels k~l)(Z)
and k~2)(Z) in the space L 2(R+), then by equalities (2.97) we have
(Kf){z) = 2-1sgn{z)[K~1)j~1)]{1 z I) +2-1[K~2)j~2)J(1 Z I). (2.100)
Since j~l)(1 z I) = j{1 z I) - j{ - I z I), j~2)(1 z I) = j{1 z I) + j{ - I z I), the function j{z) defined by relation (2.95) belongs to the space L 2(R) if and only if h(z), h(z) E L2{R+). Equation (2.100) shows that integral transform (Kj){z) is unitary in L 2(R) if and only if the transforms [K(1) j]{z) and [K(2) jl{z) are unitary in L 2 (R+). Further, we can show that the inversion formula (2.91) holds. In fact,
(Kg){z) = {k(z);g(z)} (z) = {T1[kp)(1 z I) + sgn{z)k~2)(1 z 1)];
Tl[sgn(z)[Kp)j~l)](1 z I) + [K~2)j~2)]{1 z I)]}
(2.101)
32 Chapter 2
= T 2([kP)(1 :c I); sgn(:c)[K~I) f~I»)(1 :c I)}(:c) + T 2{sgn(:c)[k12)(1 :c I);
[K~2)f~2)](I:c I)}(:c) = T 1 (sgn(:c)f~I)(I:c I) + f~2)(I:c I)) = f(:c)
due to equality (2.95). Theorem 2.8 is proved.• In general case, let the even and odd components of the functions k(:c), k(:c), :c E
R (we denote them as k1i )(:c), k1i )(:c), :c E R+, i = 1,2) satisfy the Watson condition (2.71), respectively. Then the bilateral transforms (2.90), (2.91) with k(:c) replaced by k(:c), generally speaking, are not unitary, but they are bounded on the space L 2 (R) and inverse to each other.
Let now the function k(:c) be differentiable and, moreover, k'(:c) E L 2(-a,a) for all 0 < a < 00. Then the bilateral transforms pair (2.90)-(2.91) can be written in the form
(Kf)(:c) = l.i.m.M.N +oo1: k'(:cy)f(y)dy,
(2.102)
(2.103)
where the convergence of the integrals is understood as convergence in L 2(R) and
the kernel k' (:c) is a Fourier kernel. If the kernels (kP) (:c))', (Ml)(:c))' , (k12)(:c))' ,
(k12)(:c))' are such that in the space L(R+) there exist the following representations
through the Fourier integral (E.C.Titchmarsh (1937))
f(:c) = 100
(kp)(:ct))' dt100
(kP)(yt))' f(y)dy,
f(:c) = 100
(k12)(:ct))' dt100
(k12)(yt))' f(y)dy,
(2.104)
(2.105)
then it is not difficult to show that in the space L(R) the following representation holds
f(:c) =1: k'(:ct)dt1: k'(yt)f(y)dy. (2.106)
We note that the inner integrals in relations (2.104)-(2.106) are absolutely con vergent, the outer integrals in relations (2.104)-(2.105) are understood as improper, and in relation (2.106) in the principal value sense.
Let (k11\:c))' = (kp)(:c))' = (2/11")1/2 cos(:c), (k12)(:c))' = - (k12)(:c))' = (2/1I")1/2isin(:c). Then k'(:c) = k'(:c) = (211")-1/2ei .. and we obtain the classical Fourier transform (2.87).
The kernels (kp)(:c))' = (k11)(:c))' = (2/11")1/2 cos(:c), (k12)(:c))' = (k12)(:c))' = (2/11" )1/2 sin(:c), lead to the formulae for the symmetric Hartley transforms (2.88) (2.89). New integral bilateral transforms pairs can be obtained by taking of Fourier and Watson kernels from E.C.Titchmarsh (1937). For example, making use the
Hankel transform (2.67) we obtain the bilateral symmetric unitary Hankel transform in L 2 (R)
11+ 00
g(:z:) = 2" -00 I:z:y 11/ 2 (JVl (I :z:y I) + sgn(:z:y)JI"J(I :z:y I)) f(y)dy,
f(:z:) = ~1:00
I:z:y 11/ 2 (Jv1 (I:z:y I) +sgn(:z:y)JI"J(I:z:y I))g(y)dy,
where !R(Vi) > -1, i = 1,2. The bilateral symmetric Hardy transform has the form
11+ 00
g(:z:) = 2" -00 I:z:y 11/ 2 (Yv(l :z:y D+ sgn(:z:y)Hv(1 :z:y I)) f(y)dy,
11+ 00
(2.107)
(2.108)
(2.109)
(2.110)
where Yv(z) and Hv(z) are the Neumann and Struve functions, respectively (see A.Erdelyi et al. (1953)). We note that, since the Hardy transform is not unitary in L 2(R+), but it is an automorphism in this space (E.C.Titchmarsh (1937)), so the bilateral transform (2.109) also possesses these properties.
The mentioned above Fourier kernels are the particular cases of the Kesarwani's kernels (RN.Kesarwani (1963a)) which, in turn, are the special case of the Meijer
a-function (1.46) a';,:,n (:z:I~p:n. If the conditions Pi + qi = 2(mi + ni), Pi =f q;,
~ ("Pi (i) _ "qi a(i») _ 0 ~(i») 1/2 . - 1 . ~(i») 1/2' _ :It LJj=1 aj LJj=1 fJj -,:It aj < ,J - , ... , n" ;It aj > - , J -
ni +1, ... ,Pi, !R(I3~i») > -1/2, j = 1, ... ,mi, !R(I3~i») < 1/2, j = mi +1, ... , q;, i = 1,2 hold, then the Kesarwani's kernels generate the bilateral symmetric automorphic transform in the space L 2(R) as follows
(2.111)
(2.112)
In particular, if Pi = 2ni, qi = 2mi, mi =f ni, ay) E R, a~2ni = _a~i), j =
1, ... ,ni; l3~i) E R, 13~2mi = -13~i), j = 1, ... ,mi, i = 1,2, then the bilateral transform (2.111) is unitary in the space L2(R).
34 Chapter 2
2.5 Multidimensional Watson transforms
In this last section we demonstrate the generalization of the one-dimensional Watson transforms to multidimensional case due to Nguyen Thanh Hai et ai. (1992).
We borrow heavily from the results of Yu.A.Brychkov et al. (1992), introduced in Vu Kim Tuan and Nguyen Thanh Hai (1991).
Definition 2.3 Let Sl = Il;=l Sj for s E cn; aj = {Sj E C, ~(Sj) = 1/2}, a = al x ... x an, and let a function F(s) defined in a satisfy condition (2.7)
F(s)F(l- s) = 1, I F(s) 1= 1. (2.113)
(2.114)
Then the function k(x) : R+ --t R is called the n-dimensional Watson kernel if
Xl 1 F(s) -s k(x) = (21l"i)n q (1 _ S)l X ds,
h b fl' fl/2+iN ( . ) fl/2+iNwere y Jq we mean .z.m·N-+oo Jt/2-iN ... n tzmes ... Jt/2-iN .
It is evident that if n = 1, we immediately get the Watson kernel (2.68). By using the procedure discussed in Theorems 2.6, 2.7 and Corollaries 2.3, 2.4
for one-dimensional case, it is not difficult to prove the following theorem (see Yu.A.Brychkovet al. (1992»:
Theorem 2.9 Let x, y E Rn, x 0 Y = (XlYll"" xnYn) be a coordinate product of the vectors x and Y, D~ = a.}.':axn and let the real function k(x) be an n-dimensional Watson kernel (2.114). Then the following n-dimensional transform
(Kf)(x)=g(x)=D~r k(xoY)f(y)d~ JR+ Y
is unitary in L 2(R+) and its inversion has the symmetric form
(2.115)
(Kg)(x) = f(x) = D~ r k(x 0 y)g(y) d;. (2.116)JR'.j. y
Remark 2.2 The transforms (2.115), (2.116) are called direct and znverse n dimensional Watson transforms, respectively.
Further, general formulae like (2.115), (2.116) are satisfied if and only if the complex value kernel k satisfies the conditions
1'.j. k(x 0 u)k(y 0 u)u-2du = II min(xj,Yj)' (2.117)
r k(x 0 u)k(y 0 u)u-2du = IT min(xj, Yj), (2.118) JR'.j. j=l
where u-2 = Il;=l uj2. As it was shown in Yu.A.Brychkov et al. (1992), this condition is equivalent to Definition 2.3.
Remark 2.3 If the functions k~j\l:j), j = 1, ... ,n are one-dimensional Watson kernels (2.68), then the function k(z) = TIi=l k~j)(zj) is an n-dimensional degenerate Watson kernel.
We begin with the study of nondegenerate multidimensional Watson transforms when the corresponding Watson kernels depend of minimum of arguments.
Let us prove the following
Theorem 2.10 Let min(z 0 y) = min(zlYt> ... , ZnYn) and let the function h(zl/n) be the one-dimensional Watson kernel (2.68). Then the transform
g(z) == (Wminf)(z) = D; f h(min(z 0 y))f(y) d~ JR+ Y
is unitary in L2(R") and has the symmetric inversion formula
f(z) = (WminY)(z) = D; f h(min(z 0 y))g(y) d~. JR+ Y
(2.119)
(2.120)
(2.121)
Then
Proof. Using Theorem 2.9, it is sufficient to show that the function h(min(z 0
1)) = h(min(zt> ... , Zn)) is the n-dimensional Watson kernel (2.114). Thus, we must first prove
Lemma 2.1 Let Mn{f(Z)j 8} be the n-dimensional Mellin transform (8ee Yu.A. Brychkov et al. (1992)) of the function f(z) at the point 8 E en
M n{f(Z);8} = /*(s) = f f(z)z·-ldz. JR+
8·1 Mn{h(min(z 0 1)); -8} = -1M 1{h(z); -8 .1} (2.122)
8
(8ee (1.49)), where 8 E U, 8·1 = L:i=18j and z- lh(zl/n) E L2(R+).
Proof of Lemma 2.1. Note that the function z-l h(zl/n) belongs to L2(R+) if and only if z-(n+1)/2h(z) E L2(R+). Hence from the shift property (1.58) for the Mellin transform, we have
Mdz-(n+1)/2h(z); r} = M 1{h(z); r - (n + 1)/2}, (2.123)
where 3?(r) = 1/2. Since 8 E U, 3?(8 . 1) = n/2. It is clear that Mdh(z); -8' I} exists when 8 E u. Define Ar. = {z E R+, zr. = min(z 0 I)}. Since 8 E u, we have
8r. = IMdh(Z)j -8 .1}.
36 Chapter 2
From equality (2.124) and relationship U~=l Ale = R+. (2.122) follows. This proves Lemma 2.1. •
Since h(zl/n) is the Watson kernel (2.68), z-l h(zl/n) E L2(R+), along with equality (2.68) imply
(2.125)
where the function n(T) satisfies conditions (2.113) on the line ~(T) = 1/2. The substitution of variables in (2.121) gives
(2.126)
for ~(T) = 1/2. If T = .;: , then n(1 - T) = n - s . 1 = (1 - s) . 1. Hence
n(~) = (1 - s) . l' s E 0'.
On the other hand, by using Lemma 2.1, we have the relationships
M n{:c-1 h(min(:c 01)); s} = Mn{h(min(:c 01)); -(1- s)}
(1-s).1 n(~) n(~) (l-s)1 (1-s).1 = (I-S)l ,sEO'.
(2.127)
(2.128)
The Mellin transform inversion formula of the type (1.56) in multidimensional case gives
(2.129)
If F(s) = n ('~1), then it is easily seen that condition (2.113) holds for the function F(s). In accordance with Definition 2.3 we find that h(min(:col)) is an n-dimensional Watson kernel. Applying Theorem 2.9 we complete the proof of Theorem 2.10.•
As we know, the function k(:c) = (2/'11-)1/2 sin(:c) is an one-dimensional Watson kernel. This provides the following corollary of Theorem 2.10
Corollary 2.5 The n-dimensional transform
g(:c) = (2/1r?/2D; [ sin([min(:c 0 y)t)f(y) d~ JR+ Y
is unitary in L 2 (R+.) and has the symmetric inversion formula
(2.130)
(2.131)
(2.132)
Theorem 2.1 shows that anyone-dimensional Watson kernel mentioned in this chapter will provide such an example corresponding to the kernel in (2.119). Hence the functions
1 11 + znI 1(2/7r)1/2(1 - cos(zn)), -In -- , -In 11 _ z2nl 7r 1 - zn 7r
provide examples of the kernel h(z) in (2.119), (2.120). Now we generalize the results of Section 2.4 to the multidimensional case. We
obtain an analogue of Theorem 2.8 for the Watson transform in space L2(Rn), i.e., we find the necessary and sufficient conditions on the kernels k(:z:) : Rn -t R for which n-dimensional transform
1 dy (KJ)(:z:) = g(:z:) = D~ k(:z: 0 y)f(Y)l
R+ Y
is unitary in L 2 (Rn) and its inversion has the symmetric form
h dy (Kg)(:z:) = f(:z:) = D~ k(:z: 0 y)g(Y)l
R+ Y
in the following
Theorem 2.11 The n-dimensional integral transform (2.133) is unitary in space L 2(Rn) and the symmetric inversion formula (2.134) holds if and only if k possesses the following representation
2"
(2.135)
where the function kj(:z:) is an n-dimensional Watson kernel in R+ (see Definition 2.3),
kj(l:z: I) = kj(1 :Z:1 I,·· ., I :Z:n I), A;(:Z:) = sgn { II :z:p} . (2.136) pEJj
Here J j , j = 1,2, ... ,2n is subset of the set 1= {1,2, ... ,n}, i.e., Jj E 21 and
J1 = 0, sgn{nPEJ l :z:p} = 1, J2" = I.
Proof. We note that in the case n = 1, from relation (2.135) and assertion of Theorem 2.8 we immediately get representation (2.92). For the general case, we divide the proof of Theorem 2.11 into several steps.
Lemma 2.2 Any function k(:z:) : Rn -t R has the representation
2"
(2.137)
where Aj(:Z:) and I:z: I are defined by relation (2.136) and the functions k;(l :z: I) map R+ into R.
38 Chapter 2
Proof. The case n = 1 is given in formula (2.92). When n = 2, we have
k(Xl,X2) = Tl[kl(l Xl I,X2) + sgn(xl)k2(1 Xl I,X2)]
= T 2[ku (1 xII.I X2 I) + sgn(x2)k12 (1 Xl I, I X2 I)
+ sgn(xt}k2l (1 Xl I, I X2 I) + sgn(xlx2)k22 (1 Xl I, I X2 DJ·
(2.138)
The same considerations can be used for the case n > 2.•
Lemma 2.3 We have the following relationship:
(2.139)
This fact follows from the evenness of the integrand (see the definition of Aj(X) in relation (2.136». Now, defining
we can represent f(x) (see relation (2.135» in the form
2" 2"
f(x) = Tn L !i(1 X I)Aj(X) = z-nA2"(X) L !i(1 X I)Aj(X). j=l j=l
Thus by Lemmas 2.2 and 2.3, we have
(Kf)(x) = {k(x)j f(x)} (x)
2" 2"
= Z-2n L L {Ap(x)kp(1 X I)j A2,,(X)A,(x)f,(1 X I)}(x) p=l '=1
(2.140)
(2.141 )
2"
= 2-n L Aj(x)(Kt!i)(1 X I), j=l
where the transform Kt is defined on the space L 2(R+.). Using arguments similar to the case n = 1 (see the Theorem 2.8), we obtain the statement of Theorem 2.11. Theorem 2.11 is proved.•
Remark 2.4 Let the kernel k(x) be differentiable and, moreover,
.6.: = [-a, a] X .•. X [-a, a], ".. n times
for all 0 < a < +00. Then the Watson transforms (2.133), (2.134) can be written in the form
(Kf)(z) = l.i.m.M,N_+ooI: k'(z 0 y)f(y)dy, (2.142)
(Kg)(z) = l.i.m.M,N_+ooI: k'(z 0 y)g(y)dy, (2.143)
where convergence of the integral is understood in L2 (Rn) and the kernel k' (z) is a Fourier kernel.
The two-dimensional analog of the Hartley transforms pair (2.88)- (2.89) can be written as the following Fourier kernel
k'(Zl' Z2) = ~ [sin(zt} sin(z2) + sin(zt} COS(Z2) + sin(z2) (2.144) 2v211"
X COS(Zl) + COS(Zl) COS(Z2)]'
The Watson transforms (2.133)-(2.134) in the two-dimensional case can be writ ten in the form
(2.145)
(2.146)
(2.147)
where
k(Zl' Z2) = T 2[k1(1 Zl I, IZ21) + sgn(zl)k2(1 Zl I, IZ2 I)
+ sgn(z2)k3(1 Zl I, I Z2 I) + sgn(zl z2)k4 (1 Zl I, I Z2 I)],
and the kernels kj (Zl' Z2), j = 1,2,3,4 are two-dimensional Watson kernels in R~ . They can be degenerate, i.e., representable as a product of two one-dimensional Wat son kernels (2.68) of variables Zl and X2 (see the Remark 2.2). In this case, Theorem 2.10 provides new examples of two-dimensional kernels of the form k(min(z~,x~)).
For instance, from relation (2.132) we obtain the following two-dimensional Watson kernels
(2/11") sin(zl) Sill(X2), (2/11") sin(xl)(1 - COS(Z2)), (2.148)
21/2/11"3/2 sin(xl) In 11 - x~ I, 21/2/11"3/2(1 - COS(Xl)) In I~ ~ :: I, (2/1I"?/2sin(min(x~,z~)), (2/11")1/2(1_ cos(min(x~,x~)),
/ I . (4 4 I / 11 + min( x~, xD I1 11" In 1 - mill Xl' x 2 ) , 1 11" In . (2 2) , 1- mill Zl,x2
40
Jmin(x~, xDJv(min(x~, xm·
H- and G-transforms
In this chapter, we consider Mellin convolution type transform (2.1) with Fox's H function (1.50) and Meijer's G-function (1.46) as kernels. This transform in various forms and spaces offunctions was considered by C.Fox (1961, 1971), R.N.Kesarwani (1959) -(1971), V.KKapoor and S.Masood (1968), KC.Gupta and P.KMittal (1970, 1971), S.L.Kalla (1969 a,b), C.M.Joshi and M.L.Prajapat (1976), H.M.Srivastava (1972), H.M.Srivastava and R.G.Buschman (1973, 1976, 1977, 1992), R.Singh (1970), R.G.Buschman and H.M.Srivastava (1975), P.G.Rooney (1973, 1983), Vu Kim Tuan (1987), S.G.Samko et al. (1987), Nguyen Thanh Hai and S.B.Yakubovich (1992), A.A.Kilbas et al. (1993).
3.1 Mellin convolution type transform with Fox's H-function as a kernel
First we consider a H-transform of the following form
in the special space of functions M;'~(L). As it will be showed below, this space is very convenient for the studies of the transform (3.1).
Definition 3.1 Let C, IE R be such that
2sgn(c) + sgn(,) ~ O. (3.2)
We will denote by M;'~(L) the space of functions f(z), z > 0, representable by the inverse Mellin transform (1.56) of the function /*(s), where
I*(s) Is 17 eweI9(-)1 E L(O'), 0' = {s E C : ~(s) = 1/2}.
Note that /*(s) 1 s 17 eweI9(-)1 E L(O') iff /*(s) 1 s 17 e,,"el-I E L(O') and in this case, the integral (1.56) converges if c > 0, I E R, or c = 0, I ~ 0, which is equivalent to condition (3.2).
41
42
The space M;'~(L) is a Banach space with the norm
Chapter 3
(3.3)
It is obvious that the space M;'~(L) in the case c = 0, '1 = 0 coincides with the space M-1(L). The following theorem holds
Theorem 3.1 For the family of spaces M;'~(L) the inclusion
holds if and only if 2sgn(C1 - c) + sgn('"(1 - '1) ~ o.
Proof. Suppose, that inequality (3.5) holds. Then from relation (3.3) we have
=~1e1l"cd9 (')1 I S'Y1 J*(s )e,,"(c-cdI9(')ls 'Y-'Y1 ds I 211" iT
5, 0- 2 1 1e,,"cd9 (')1 I S'Y1 J*(s)ds 1= 0llfIlM - 1 (L)' W u Cl~l
where the constant 0 is defined by the following relation
o =sup (e1l"(c-cdI9(')1 I s 1'1'-'1'1) < +00. • EiT
(3.4)
(3.5)
(3.6)
Now suppose, that inclusion (3.4) holds valid. Let us consider the function /*(s) = e-1I"cd9 (')1 Is 1-'1'1-1-., € > O. Then f(z) E M';~'Y1(L). From inclusion (3.4) we have the following estimate
= ~1e1l"(c-cdI9(.)1 I s'Y-'Y1-1-<ds 1< +00, € > O. 211" CT
From relation (3.7) we obtain
2sgn(C1 - c) + sgn('"(1 - '1 + €) > 0, € > 0, (3.8)
which is equivalent to condition (3.5). This completes the proof of Theorem 3.1. •
Remark 3.1 We found that M';~'Y1 (L) == M;'~(L) if and only ifc1 = c and '11 = '1. We have also the following inclusions
where '1 E R, '11 E R, € > 0, €1 > O.
H- and G-transforms 43
From the last property of the space M;'~(L) it is not difficult to obtain the following result.
Corollary 3.1 Let inequality (3.5) hold for some pairs (CI,')'I) and (C2,')'2). Then
M;'~,,(I (L) U M~~'Y2(L) == M;'~(L),
where the pair (c, ')') is defined as follows
{
(c, ')') = (C2, ')'2), C2 < CI, (cI,min(-Y1l')'2)), Cl = C2.
Theorem 3.2 Let f(x) E M;'~(L), g(i) E M;'~(L). Then the function
h(x) = X I
(3.10)
(3.11)
Proof. According to Definition (3.1) of the space M;'~(L), we can represent the function (3.10) in the following form
h(x) = (;~~;2I,I. f*(s)g*(t)x-·-tdsdt,
where 0". X O"t = {(s,t) E C2, ~(s) = ~(t) = 1/2}. Using the change of variables r = s + t - 1/2, t = t, we can rewrite relation
(3.11) as follows
where 0".,. = {r E C, ~(r) = 1/2} and
F(r) = -2 1 .1 f*(r - t + 1/2)g*(t)dt, r E 0".,.. 1I'l u,
(3.12)
(3.13)
According to Definition 3.1, the function h(x) E M;'~I (L), if F(r) I r '''(I xe'l*1 E L(O".,.). Using representation (3.13), we obtain the inequality
IT ewel.,.1 Ir pI I F(r)dr I
~ ~11 ewel.+t-1/21 Is + t - 1/2 1"(11 f*(s)g*(t)dsdt I· 211' u, u.
Since f(x) E M;'~(L), g(x) E M;'~(L), we have the following inclusions
44
Hence, it is evident that the last double integral converges if
Chapter 3
sup e1fc(lo+t-l/21-1·1-ltD I s + t - 1/2 '''11/ S r"l I t r"l < 00. (3.14) (••t)Eu.xu.
We have for c;::: 0 and (s,t) E u. X Ut
c(1 s + t - 1/2 I - I s I - 1t I) ~ c/2.
Let first 1 ;::: o. Then 11 = 1 and we obtain for s E u., t E Ut
(3.15)
Is + t - 1/21"1'1 S 1-"1' t 1-"1=' S + t - 1/2 1"1' s 1-"1' t 1-"1 (3.16)
< 0 1 I ! + ! 1"1< O2 < 00. s t
If 1 < 0, then 11 = 21 and we have for s E u., t E Ut
t s < 0 3 11 + ~ 12"1' t 1"1< 0 4 < 00, 1s 1;:::1 t I;
Is + t - 1/2 1"111 s 1-"1' t 1-"1=1 s + t - 1/2 12"11 s r"ll t 1-"1 (3.18) s t
< 0 5 11 + t 12"11 ~ '''1< 0 6 < 00, It ';:::1 s I .
It follows from estimates (3.15)-(3.18), that inequality (3.14) holds under the con ditions of Theorem 3.2 and, consequently, this theorem is proved.•
Now we consider the H-transform (3.1) in the space M;'~(L).
Theorem 3.3 Letf(z) E M;'~(L) and
~((3j) +bj /2 > 0, 1 ~ j ~ mj 1 - ~(aj) - aj/2 > 0, 1 ~ j ~ n. (3.19)
Then the H - transform (3.1) exists under the following conditions
2sgn(lI:) + sgn(p, - 1) > OJ
(3.20)
(3.21)
where
(3.22)
(
p q) I(P q) P, = ~ ~ aj - ~ (3j + 2: ~ aj - ~ bj - P ; q
and under these conditions we have
g(z) = [Hfl(z) E M;~t<."I+IJ(L).
(3.23)
(3.24)
45
R(Oj) +aj/2 > 0, n +1 ~ j ~ p; 1 - R(f3j) - bj /2 > 0, m +1 ~ j ~ q, (3.25)
then the H - transform is one-to-one from the space M;,;(L) onto the space M -;';,.,7+,. (L) and its inverse has the following form:
a) In the case K, = 0, I" > 1
f( ) - [H A
]( ) - ~ Ie100
H q- m,p-n+l ( I(Up+l, ap+t}) ( )dx - 9 x - d Ie x p+l,q+l xu (a 1, ) 9 u u, x 0 fJq+l, q+l
(3.26)
(3.27)
(3.28)
where kEN, k> 1"+ 1, (uP+l' ap+t) = {(O,l), (1- 0i - ai, ai)~+l' (1- 0i - ai, ai)~}, (,8q+l,1,q+t) = {(1- f3i - bi,bi)~+l' (1- f3i - bi,bi)'{', (-k, I)}.
b) In the case K, > ° f( x) = [ifg](x) = lim k
le x le/ r (rx(r+l)/r.:l:.-.) Ie
Ie_oo (k - I)! dx
100 Hq-m+l,p-n (kr I (up, ap) ) ( )dx p q+l xu A A 9 u u, 0' (f3q+l' bq+l)
wherer E N, r > 2K" (up,ap) = {(1-0i-ai,ai)~+l,(1-0i-ai,ai)i}, (,8q+l,1,q+t) = {(O, r), (1 - f3i - bi , bi)~+l' (1 - f3i - bi , bi )'{'}.
Proof. First we will prove, that under the conditions (3.19), (3.20) the kernel of transform (3.1) belongs to the space K, of kernels. In fact, under these conditions we have the following representation of the kernel
k(x) = H;:t (xl (op,a p )) = ~lcp(s)x-'ds,
(f3q, bq) 2n u
where OJ E C, 1 ~ j ~ p, f3j E C, 1 ~ j ~ q, aj > 0, 1 ~ j ~ p, bj > 0, 1 ~ j ~ q, and the function <I>(s) is determined by the formula (1.51). Using formula (1.12) we obtain the following estimates for the function (1.51) with the constant R(s)
Al I S 1-"-(~(')-1/2)B e-...,.I.I ~I <I>(s) I~ A2 I s 1-,.-(~(·)-1/2)B e-...,.I.I, (3.29)
where Al > 0, A 2 > 0, B are some constants, K, and I" are determined by relations (3.22), (3.23). Note that the first estimate in formula (3.29) is valid under conditions (3.19), (3.25) and the second holds under condition (3.19). From the estimates (3.29) we find that <I>(s) E L(O") and from relation (1.51), <1>(1/2 + it) has the bounded variation in the neibourhood of any point t E R. We can apply Theorem 1.2 now to find that
*

the hypergeometric approach to integral transforms and convolutions

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