JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 220, 508]527Ž. 1998 ARTICLE NO. AY975836 The Weierstrass Transform for a Class of Generalized Functions V. Karunakaran and T. VenugopalU School of Mathematics, Madurai Kamaraj Uni¨ersity, Madurai, 625 021, India View metadata, citation and similar papers at core.ac.uk brought to you by CORE Submitted by John Hor¨ath provided by Elsevier - Publisher Connector Received February 18, 1997 The classical theory of the Weierstrass transform is extended to a generalized function space which is the dual of a testing function space consisting of purely entire functions with certain growth conditions developed by Kenneth B. Howell. An inversion formula and characterizations for this transform are obtained. A comparative study with the existing literature is also undertaken. Q 1998 Academic Press The conventional Weierstrass transform of a suitably restricted function ftŽ.on the real axis R is defined as 1 2 HfteŽ.yŽ zyt. r4 dt, '4p R where z is a complex variableŽ see, for example,wx 3. This transform arises naturally in problems involving the heat equation for one dimensional flow. Zemanianwx 12 defined and investigated the Weierstrass transform of a certain class of generalized functions which are duals of the so-called testing function spaces Wa, b and W Ž.a, b . The inversion formulas were also obtained. The intrinsic connection between the Weierstrass and the Laplace transforms is also brought out in Theorem 7.2.1wx 12, p. 208 , Theorem 7.2.2wx 12, p. 209 , and Theorem 7.3.1 wx 12, p. 211 . Further, necessary and sufficient conditions for a function FzŽ.to be the Weier- strass transform of a generalized function are also obtainedŽ seew 12, * Research of this author is supported by a senior research fellowship from CSIR, India. 508 0022-247Xr98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved. GENERALIZED WEIERSTRASS TRANSFORM 509 Theorem 7.3.5, p. 213x. On the other hand the Weierstrass transform of bounded functions, L p-functions, and certain other functions with pre- scribed growth conditions are all characterized by Hirshman and Widder wx3 . For other types of Weierstrass transforms on single or multidimen- sional spaces we refer towx 2 . We first observe that the class S X of tempered distributions is con- X tained in W Ž.y2b, y2a . Therefore the Weierstrass transform theory can be made applicable to S X. The theory of Laplace transform can be easily applied to S XXand in fact it is generalized to the space L Ž.a, b Ž.when a - b defined by Zemanianwx 12 . However, there is only an X X isomorphism between W Ž.y2b, y2a and L Ž.a, b . In this sense the Weierstrass transform is not directly applicable to the generalized function space L XŽ.a, b when a - b. Also no such theory is available for X X WŽ.y2b,y2aand L Ž.a, b when a ) b. In this paper we start with the testing function space G and the generalized function space GX, the dual of G, introduced and developed in a sequence of papers by Kenneth B. Howellwx 4]8 . For suitable real numbers a and b we shall show that L XŽ.a, b can be identified as a sub-space of GX. We introduce the Weierstrass transform on GX, obtain an inversion formula, and also characterize the Weierstrass transform of elements from both G and GX. In contrast to the generalized function spaces which are duals of testing function spaces consisting of smooth complex valued functions of a single real variable, the space GX is the dual of the testing function space G which consists of entire functions with certain growth conditions. One may observe that the Fourier transform theory is applicable to all of GX Žsee wx4]8.Ž.Ž. , and the convolution transform theory with kernel 1r2 sech zr2 X XX X is also applicable to a certain sub-space E ddŽ S ; E .of G Žseewx 10. 1. PRELIMINARIES In this section we shall state the concepts and results which will be needed in the sequel. For details and proofs we refer towx 4]8. DEFINITION 1.1. The space G consists of all entire functions fŽ.z of one complex variable z such that for every a ) 0 a<Re z< 55fassup e <<f Ž.z - `, zgBa where Ba s Äx q iy s z g C : <<y F a4. 510 KARUNAKARAN AND VENUGOPAL DEFINITION 1.2. The space G c consists of all entire functions fŽ.z of one complex variable z such that for every a ) 0 there is a corresponding g G 0 such that sup eyg <Re z< <<f Ž.z - `. zgBa Ä55f 4 Gcan be given a Frechet topology with the multi-norm a a G 0 , i.e., fnnªfin G m 55f y f a ª 0as nª`;a)0. For each f g G its Fourier transform is defined as 1 itz FŽ.Ž.f z sfà Ž.z s Hf Ž.tey dt. '2pR THEOREM 1.3. The Fourier transform F is a continuous, linear, one-to-one mapping of G onto G with a continuous in¨erse. Moreo¨er, for e¨ery a ) 0, b ) 0, and f g G, 12' 55FŽ.faF 55faqb. ž/b'p THEOREM 1.4. For each c g G, the mapping f ª f )c is a continuous linear mapping from G into G where 1 Ž.Ž.f )c z s Hf Ž.Ž.t c z y tdt '2pR Ž. with 55f )c a F 2rb'2pf5555aa cqbfor any a ) 0, b ) 0. Moreo¨er, the con¨olution product on G is commutati¨e and associati¨e and for any f, c g G, m g N we ha¨e Žm.Žm.Žm. Ž.i Žf)c . sf)csf)c Ž.ii F Žf )c .s F Ž.Ž.f F c and F Žfc .s F Ž.f ) F Ž.c . Let us denote the space of continuous linear functionals on G by GX. GX will be given the weakU topologyŽ seewx 11, 3.14, p. 66. This is a locally convex vector topology on GX and compact sets of GX in this topology will U U be called Weak compact sets. Moreover fn ª f in the weak topology of X Gif and only if fnŽ.f ª f Ž.f for every f g G. X THEOREM 1.5. For each f g G there are finite positi¨e constants C and a such that<²: f, f <55F C f a for all f g G. X The Fourier transform F wxf of an f g G is defined to be an element of X Gby ²F wxf , f:²s f, F Ž.:f for f g G. GENERALIZED WEIERSTRASS TRANSFORM 511 X THEOREM 1.6. F is continuous, linear, one-to-one mapping from G onto X G with continuous in¨erse. X If G g G , c g G the convolution of G with c , G)c , is the function ÏÏÏ given by Ž.Ž.Ž.G)f z s F tczzwhere tc Ž.Žw sc zyw .with c Ž.z s cŽ.yz. X c THEOREM 1.7. If G g G , c g G then G)c g G . Moreo¨er, Ž.i ŽG)f .)csG) Žf)c . Ž.ii F ŽG)c .s F ŽG .F Žc .and F Žc G .s F ŽG .) F Žc ., where Ž.Ž.Ž.cGfsGcf for f g G. 2. THE WEIERSTRASS TRANSFORM X DEFINITION 2.1. If f g G then the Weierstrass transform fÄ of f is defined to be Ä fwŽ.s²:fz Ž.,kw Žyz,1 .s'2p Žf)kw1 .Ž. Ž wgC ., 2 yzr4t where kŽ.z, t s kt Ž.Žz s 1r '4p t .e for t ) 0, z g C ŽOne can easily verify that kwŽ.yz, 1 , as a function of z, belongs to G for every w g C.. The following proposition is an immediate consequence of Theorem 1.7; X c PROPOSITION 2.2. If f g G then its Weierstrass transform fÄ belongs to G . We shall now obtain the inversion formula for the Weierstrass trans- form of elements in G c and using this, we shall also deduce the inversion formula for the Weierstrass transform of elements in GX. Though the classical proof does not seem to be adaptable to this case, we would like to remark that the proof essentially makes use of the continuity of the Fourier transform and its inverse from G onto G. THEOREM 2.3Ž Inversion for the Weierstrass transform of elements in c c G.Ž.If fÄ is the Weierstrass transform of f g G then for z g C, f z.s lim H kŽ.Ž.j iz, t fÄ ij dj. t ª 1y R q c Proof. Since f g G there are positive constants, M and a, such that a<j< <fjŽ.<FMe for all j g R. By a straightforward calculation we obtain <<<fÄŽ.xqiy s ²:fj Ž.Ž,kxqiy y j ,1 .< 1 Žxiy j.24 s Heyqy rfjŽ.dj '4p R y2 4a<x< Fere N, 512 KARUNAKARAN AND VENUGOPAL 2 ys r4 a < s< where N s Ž.1r '4p HR eMedswhich is clearly finite. From this it immediately follows that HR kŽ.Ž.j q iz, t fÄ ij dj exists whenever 0 - t - 1. To prove the inversion formula, let c Ž.z, t s Hk Žj q iz, t .Ž.fÄ ij dj .