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. ᮊ 1998 Academic Press
The conventional Weierstrass transform of a suitably restricted function ftŽ.on the real axis ޒ is defined as
1 2 HfteŽ.yŽ zyt. r4 dt, '4 ޒ
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.␣,  . 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 ᮊ 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.y2, y2␣ . 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.␣,  Ž.when ␣ -  defined by Zemanianwx 12 . However, there is only an X X isomorphism between W Ž.y2, y2␣ and L Ž.␣,  . In this sense the Weierstrass transform is not directly applicable to the generalized function space L XŽ.␣,  when ␣ - . Also no such theory is available for X X WŽ.y2,y2␣and L Ž.␣,  when ␣ ) . 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 ␣ and  we shall show that L XŽ.␣,  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 ޅ ddŽ S ; ޅ .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 Ž.z of one complex variable z such that for every ␣ ) 0
␣ where B␣ s Äx q iy s z g ރ : < DEFINITION 1.2. The space G c consists of all entire functions Ž.z of one complex variable z such that for every ␣ ) 0 there is a corresponding ␥ G 0 such that sup ey␥ Ä55 4 Gcan be given a Frechet´ topology with the multi-norm ␣ ␣ G 0 , i.e., nnªin G m 55 y ␣ ª 0as nªϱ᭙␣)0. For each g G its Fourier transform is defined as 1 itz FŽ.Ž. z sˆ Ž.z s H Ž.tey dt. '2ޒ 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 ␣ ) 0,  ) 0, and g G, 12' 55FŽ.␣F 55␣q. ž/' THEOREM 1.4. For each g G, the mapping ª ) is a continuous linear mapping from G into G where 1 Ž.Ž. ) z s H Ž.Ž.t z y tdt '2ޒ Ž. with 55 ) ␣ F 2r'25555␣␣ qfor any ␣ ) 0,  ) 0. Moreo¨er, the con¨olution product on G is commutati¨e and associati¨e and for any , g G, m g ގ we ha¨e Žm.Žm.Žm. Ž.i Ž) . s)s) Ž.ii F Ž ) .s F Ž.Ž. F and F Ž .s F Ž. ) F Ž. . 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. ª f Ž. for every g G. X THEOREM 1.5. For each f g G there are finite positi¨e constants C and ␣ such that<²: f, <55F C ␣ for all g G. X The Fourier transform F wxf of an f g G is defined to be an element of X Gby ²F wxf , :²s f, F Ž.: for 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 G the convolution of ⌫ with , ⌫) , is the function ˇˇˇ given by Ž.Ž.Ž.⌫) z s F zzwhere Ž.Žw s zyw .with Ž.z s Ž.yz. X c THEOREM 1.7. If ⌫ g G , g G then ⌫) g G . Moreo¨er, Ž.i Ž⌫) .)s⌫) Ž) . Ž.ii F Ž⌫) .s F Ž⌫ .F Ž .and F Ž ⌫ .s F Ž⌫ .) F Ž ., where Ž.Ž.Ž.⌫s⌫ for g G. 2. THE WEIERSTRASS TRANSFORM X DEFINITION 2.1. If f g G then the Weierstrass transform f˜ of f is defined to be ˜ fwŽ.s²:fz Ž.,kw Žyz,1 .s'2 Žf)kw1 .Ž. Ž wgރ ., 2 yr4t where kŽ., t s kt Ž.Ž s 1r '4 t .e for t ) 0, g ރ ŽOne can easily verify that kwŽ.yz, 1 , as a function of z, belongs to G for every w g ރ.. The following proposition is an immediate consequence of Theorem 1.7; X c PROPOSITION 2.2. If f g G then its Weierstrass transform f˜ 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 ˜ is the Weierstrass transform of g G then for z g ރ, z.s lim H kŽ.Ž. iz, t ˜ i d. t ª 1y ޒ q c Proof. Since g G there are positive constants, M and ␣, such that ␣<< <Ž. y2 4␣ 2 ys r4 ␣ < s< where N s Ž.1r '4 Hޒ eMedswhich is clearly finite. From this it immediately follows that Hޒ kŽ.Ž. q iz, t ˜ i d exists whenever 0 - t - 1. To prove the inversion formula, let Ž.z, t s Hk Ž q iz, t .Ž.˜ i d . ޒ We need to show that, as t ª 1 y , Ž.z, t ª Ž.z for all z g ރ. Writing in all detail, Ž.z, t s Hk Ž q iz, t .Ž.˜ i d ޒ 1 Žiz.224t Ži s. 4 s HHeeyqr y y rŽ.sdsd. 4't ޒޒ It is not hard to show that, so long as 0 - t - 1, the above integrand is an analytic function of both s and as well as being an absolutely integrable function on ޒ 2. This allows us to use the ‘‘change of variables’’ Ž.actually, a change of variables and Cauchy’s theorem and the interchang- ing of the order of integration employed in the next set of computations: 1 Žiz.224t Ži s. 4 Ž.z,t s HHeeyqr y y rŽ.sdsd 4't ޒޒ 1 w224tŽiw . 4 s HHeeyryy¨ r Ž.¨qzd¨dw. 4't ޒޒ This last integral is equal to 111t 242yi HHŽ.¨qzey¨rexp y ed¨d¨. ''2tޒޒž/2ž/t Noting that the inner integral on the last line is the inverse Fourier transform of a Gaussian, we easily obtain 112 y¨r4 Ž.z,tsH Ž¨qze .␥ Ž.¨ d¨s Ž.⌫)␥Ž.0, ''2ttޒ where 221 y¨r4yŽ␥¨.r4 ⌫Ž.¨s Ž¨qze . , ␥Ž.¨ s ␥e , and '2 t ␥s . (1yt GENERALIZED WEIERSTRASS TRANSFORM 513 By straightforward analysis or simply appealing to, say,w 8, Lemma 2.3, p. 570x , we immediately obtain 0 2 4 lim Ž.z, t s Ž0 q ze .y r s Ž.z. tª1y The following Lemma 2.4 and Corollary 2.5 can be seen to be particular z 2 4 cases of Lemma 2.3 inwx 8, p. 570 taking Ž.z s Ž1r '2 .ey r and ␥sŽ.1r't. LEMMA 2.4. If ktŽ. s k Ž, t . and if g G then q '2Ž.)ktª in G as t ª 0. XXq COROLLARY 2.5. If f g G then '2 Ž.f ) kt ª finG as t ª 0. X THEOREM 2.6Ž. Inversion Theorem . If f g G and if f˜ is the Weierstrass transform of f then ²: ˜ f,s'2lim lim HkyŽ.Ž.Ž.qix, tfŽ.)kiydyr ,x. r0¦;t1 ªªyޒ Proof. We observe that using properties of convolution ˜˜ f)krsŽ.''2f)k1)krrs2Ž.f)k)k1s Ž.f)kr. ˜ c Now Ž.f ) krrbeing the Weierstrass transform of f ) k belongs to G by Theorem 1.7. Applying Theorem 2.3 we get ˜ Ž.Ž.f ) kxrrslim Hky Žqix, tf .Ž.Ž.)kiydy tª1yޒ ˜ slim HkyŽ.Ž.qix, tfŽ.)kiydyr . tª1yޒ Thus ²:f,s'2lim ²Ž.f ) kr, : Žby Corollary 2.5 . rª0 ˜ s'2lim lim HkyŽ.Ž.Ž.qix, tfŽ.)kiydyr ,x. r0¦;t1 ªªyޒ X COROLLARY 2.7. If f, g g G and if their respecti¨e Weierstrass transforms X f˜ and˜ g are equal pointwise, then f s g in the sense of equality in G . 514 KARUNAKARAN AND VENUGOPAL Proof. For g G by Theorem 2.6 ²: ˜ f,s'2lim lim HkyŽ.Ž.Ž.qix, tfŽ.)kiydyr ,x r0¦;t1 ªªyޒ s'2lim lim HkyŽ.Ž.Ž.qix, tgŽ.˜)kiydyr ,x r0¦;t1 ªªyޒ s²:g,. 3. CHARACTERIZATIONS OF THE WEIERSTRASS TRANSFORM In this section we shall characterize the Weierstrass transform of ele- ments of G and GX. The characterization for Weierstrass transforms of elements of G is done in the canonical way. However, in the case of GX the proof of the theorem illustrates the effective use of the Fourier X transform theory on G Ždeveloped inwx 4᎐8 and other new concepts and X results regarding G Ždeveloped inwx 9. . THEOREM 3.1Ž Characterization of the Weierstrass transform for ele- ments of G.. The conditions 24 Ž.IF Ž w . is entire and< FŽ. u q i¨ tD2 ey FwŽ.sHk Žqiw, tFi .Ž .d, ޒ are necessary and sufficient conditions that FŽ. w s ˜ Ž.w for some g G. Proof. Let us first prove thatŽ. I and Ž. II are necessary. If FwŽ.s˜ Ž.w for some g G, then it is easily checked that F satisfiesŽ. I with w u iand <Ž.< M. As a simple application of Morera’s theorem sq¨ F2 one can show that eFwytD Ž.is entire. Moreover by Fubini’s theorem and a series of straightforward calculations, tD2 ey FuŽ.sHHk Žqiu, tFi . Ž .ds ku Žy,1yt . Ž.d. ޒޒ In obtaining the last equality we have used Theorem 2.4 inw 3, Chap. VIII, p. 177x . Therefore tD2 ey FuŽ.sHku Žy,1yt . Ž.d ޒ GENERALIZED WEIERSTRASS TRANSFORM 515 for real u. Furthermore since both sides are entireŽ the last integral is in fact a convolution of two elements of G and thus is an element of G. 2 ytD Ž.' Ž .Ž. ރ eFws2 )kw1yt for all w g by the principle of analytic ' Ž.Ž. continuation. By Lemma 2.4, 2 )kw1yt converges in G to as t 1 . Thus for each ␣ ) 0 and ) 0 there exists c - 1 such that ª 2y ytD 5 eFy5␣ -Ž.᭙t)c. Thus 2 ytD 5 eF555␣Fq␣ ᭙t)c.1Ž. For t F c we observe that as 1 y t - 1 e ␣ < u< ␣ 2 yzr4 Žfor some  ) ␣ and where C12s 5e 555␣ and C s  . e␣ Thus 2 ytD 5 eF5␣FN␣ ᭙tFc.2Ž. FromŽ. 1 and Ž. 2 we obtain Ž II . . Conversely suppose thatŽ. I and Ž II . hold. By condition Ž. I , condition Ž II . 2 ŽŽ.observe that condition II in particular implies that uŽ.Ž.Ž. , t q ␦ s Hk y , tu,␦ d Ž.3 ޒ Ž1t.D2 for 0 - ␦ - 1, 0 - t - 1 y ␦, and yϱ - x - ϱ where eFyyŽ.is denoted by uŽ., t . The family ÄuŽ. , ␦ 4␦ is uniformly bounded on every compact subset of ރŽŽin fact on every strip around the real axis by II.. and thus is normal with respect to ރ and as such there exists a subsequence ÄuŽ. , ␦n 4 516 KARUNAKARAN AND VENUGOPAL converging uniformly on compact subsets to a holomorphic function Ž.z Žseewx 1, Definition 2, Sect. 5.1, p. 220, and Theorem 15, Sect. 5.4, p. 224. . This is entire andŽŽ.. as a simple application of II can be easily verified to be in G. Allowing ␦ ª 0 inŽ. 3 we get by the Lebesgue dominated convergence theorem uŽ. , t s Hޒk Ž y , t .Ž.d. The last integral can be verified to be a continuous function of t, and letting t ª 1 y in the last equality, we see that uŽ.Ž ,1ysHޒky ,1.Ž. dand as in the second part of Theorem 6.3 ofw 3, Chap. VIII, p. 187x , uŽ.Ž. ,1ysF and by principle of analytic continuation uwŽ, 1ys.Ž.Ž.Ž.Fw or Fws˜w for all w g ރ. To characterize the Weierstrass transform of elements of GX, we shall need some concepts and lemmasŽ. from Definition 3.2 to Lemma 3.4 due to Karunakaran and Kalpakamwx 9 . DEFINITION 3.2. The class ⌬, by definition, consists of all sequences Ä4␦n from G such that H␦nŽ.xdxs'2 Ž.⌬1 ޒ H<<␦nŽ.xdxFM Ž.⌬2 ޒ ␣ and the class ⌬ˆˆconsists of all Ä4␦nnwhere Ä4␦ g ⌬. LEMMA 3.3. Let Ä4␦n g ⌬. For each fixed ␣ ) 0 and ) 0 the condition ␣ < x < lim HŽ.Ž.e y 1 <<␦nxdxs0.Ž. 4 nªϱ< ␣ < x < lim HŽ.Ž.e y 1 <<␦nxdxs0.Ž. 5 nªϱޒ Proof. It suffices to prove thatŽ. 4 implies Ž. 5 for each fixed ␣ ) 0 and ␣ Hence ␣ < x < ␣ ␣ <²:Ž.nnfyf,<55FC, where Cn ª 0 as n ª ϱ. ŽNote that this in particular implies n f tends to f in the weakU sense also.. X Proof. Now since f g G , by Theorem 1.5 there are positive constants c and ␣ such that for all g G, <<55fŽ.Fc␣.6Ž. So <<<<Ž.Ž.Ž.nnfyfsfy Fc55n y ␣.7Ž. Denoting the constant function 1Ž.z s 1 by 1 we have ␣ ␣ ␣ Ž␣y. Fccn55 ,8Ž. Ž␣y. We shall now prove that cn ª 0as nªϱ. Using the properties of elements of ⌬ we get if ␦ˆˆnns Žas Ä4 ng ⌬. Ž␣y. Let R be such that for a given ) 0, eŽ ␣y . R - ,10Ž. 3M where M satisfies Ž.⌬2 . Let A and B be the subsets of B␣ consisting of all zsuch that<< Re z F R and << Re z ) R, respectively. Then in the compact z set A, e is uniformly continuous. Hence given r2 M there exists ) 0, such that itz < Now since B␣ is the disjoint union of A and B, it suffices to show that the supremum over both A and B tends to 0 as n ª ϱ. Put 1 yitz I1s H <<<<␦nŽ.te y1dt '2 < In view ofŽ. 11 and Ž⌬2 . I - .12Ž. 1 2 For z g A and < itz ␣ where K s 2rŽ1 y ey. and ) ␣. Thus by Ž.⌬3 , for large n, I - .13Ž. 2 2 GENERALIZED WEIERSTRASS TRANSFORM 519 ByŽ. 12 and Ž. 13 and the fact that ␣ -  we have 1 Ž ␣y . Take z g B. Then for sufficiently large n 1 Ž ␣y . The last inequality is obtained using Lemma 3.3 and Ž.⌬2 and by writing ␣ < t < ␣ < t < e q1as e y1q2. FromŽ.Ž. 9 , 14 , and Ž. 15 we get for large n, Ž ␣y . This proves that cnnª 0as nªϱ. Hence we have fromŽ. 8 , <Ž.Ž. f y f < Fccnn55  where c ª 0as nªϱ. X COROLLARY 3.5. For f g G , and Ä4␦nng ⌬, f )␦ ª fasnªϱ ‘‘strongly’’ in GX in the sense that for some ) 0, <²:Ž.f)␦nnyf,<55FD, where Dn ª 0 as n ª ϱ. X Proof. Let f g G and g G. Then we have <²:ˆˆˆˆ<5555ˆ Ž.f)␦nnnnyf,s¦;ž/␦fyf,FCFD. The equality holds by virtue of the definition of the Fourier transform on GX and by Theorem 1.7Ž. ii . The first inequality holds by Lemma 3.4 X since Ä4␦ˆˆn g ⌬ and f ˆg G . The last inequality holds by Theorem 1.3 for some ) 0 where Dnnis a constant multiple of C . Thus for all g G <²:Ž.f)␦nnnyf,<55FD, where D ª 0as nªϱ. 520 KARUNAKARAN AND VENUGOPAL We shall state the following three lemmas without proofs. The proofs of Theorem 5.3, Lemma 6.2, and Theorem 3.2 ofw 3, Chap. VIII, pp. 184, 185, 182x can be easily adapted to prove Lemma 3.6, Lemma 3.7, and Lemma 3.8, respectively, once we note that the crucial integrability conditions involved therein remain valid in our case also when F is assumed to have the following property: ␣ LEMMA 3.6. If Ž1 t. D 2 uŽ. , t s ey y F Ž. and ¨Ž.Ž.Ž.,tq␦sHky,tu,␦ d ޒ then Ž2. Ž.Ž.Ž.iu,tq␦and ¨ , t q ␦ , as functions of Ž. , t , are in C and u s ut and ¨ s ¨t in 0 - t F c - 1 for e¨ery c - 1. Ž.ii lim ŽŽu , t q ␦ .y ¨ Ž , t q ␦ ..s 0 for all 0 g ޒ. ª0 tª0q Ž. Ž. <Ž. ␦ <Ž. <Ž iii If h10s max -tFcu , t q and h20s max -tFc¨ , ␣2 b2 tq␦. LEMMA 3.7. For 0 - ␦ - c,0-t-cy␦,c-1, and yϱ - - ϱ, uŽ.Ž.Ž. , t q ␦ s Hk y , tu,␦ d. ޒ LEMMA 3.8. uŽ. , t s Hޒk Ž.Ž, tFqi .d and u Ž w,1ys .Fw Ž. for all w g ރ. THEOREM 3.9Ž Characterization of the Weierstrass transform for ele- ments of GX .. The conditions Ž.I F has property ŽP . 2 ytD Ž.II <²eF,:<55FM␣ Ž.0-t-1, g G for some ␣ ) 0, where, using the notation of wx3, p. 179 , tD2 ey FwŽ.sHk Žqiw, tFi .Ž .d ޒ X are necessary and sufficient conditions that FŽ. w s f˜ Ž. w for some f g G . GENERALIZED WEIERSTRASS TRANSFORM 521 Proof. We first prove that conditionsŽ. I and Ž II . are necessary. If X FwŽ.sfw˜ Ž.for some f g G , then we have with w s u q i¨, << 2 eywr4 Žz22wz. 4 s²:fzŽ.,e yr '4 22 2 Žuy¨.r4Žzy2wz.r4 FMe1<< e <<␣Ž.by Theorem 1.5 ␣ X If f g G and fnns f )␦ for some Ä4␦ ng ⌬ then by Corollary 3.5, there exists  ) 0 such that <²:Ž.fnnnyf,< <²:fn,<55FMŽ.gG.16 Ž. By an application of Fubini’s theorem we get HHfxknŽ.Žqiu, tki .Žyx,1 .ddx ޒޒ sHHfxknŽ.Žqiu, tki .Žyx,1 .dx d . ޒޒ That is, fzŽ., k Ž iu, tki .Ž z,1 .d ¦;n H q y ޒ ²: sHfznŽ.,k Žqiu, tki .Žyz,1 . d. ޒ Allowing n ª ϱ we see thatŽ as a consequence of the Lebesgue’s dominated convergence theorem which is seen to be valid usingŽ.. 16 ¦;fzŽ.,Hk Žqiu, tki .Žyz,1 .d ޒ sH²:fzŽ.,k Žqiu, tki .Žyz,1 . d. ޒ 522 KARUNAKARAN AND VENUGOPAL Now, tD2 ey FuŽ.sHk Žqiu, tFi . Ž .d ޒ sHkŽ.Ž.Ž.qiu, tfz²:,kiyz,1 d ޒ sH²:fzŽ.,k Žqiu, tki .Žyz,1 . d ޒ s²:fzŽ.,Hk Žqiu, tki .Žyz,1 .d ޒ X s²:fzŽ.,ku Žyz,1yt .gG . In obtaining the last inequality we have used Theorem 2.4 ofw 3, Chap. VIII, p. 177x . 2 ytD Ž.' Ž .Ž. ރ Therefore eFws2f)kw1yt for w g by the principle of analytic continuation. Thus, if g G, 2 ytD ˇˇ' Ž.eFyfŽ.sŽ.2Ž.f)k1ytyfŽ. 'ˇ sŽ.2Ž.Ž.f)k1yt)0yfŽ. 'ˇ s2Ž.f)Ž.Ž.k1yt)0yfŽ. ' ˇˇ sfŽ.2Ž.k1yt)yfŽ. f'2k ˇˇ sž/Ž.1yt) y ' ˇˇ FM112Ž.kyt) y  ' sM112Ž.kyt)y ' ˆˆ FM212Ž.kyt) y q 2 ˆyŽ1yt.z FM2 55q sup X observe that for z g Bq Ž1 t. z 2 Ž1 t. x 2 < ey y y 1< F MeŽ.y y q1F2M GENERALIZED WEIERSTRASS TRANSFORM 523 2 and so M sup < eyŽ1 yt. z 1< is bounded. Hence fromŽ. 17 we tzs gBq y obtain 2 ytD Ž.'2eFŽ.FMt55q Therefore for some ␣ ) 0 and for all g G, 2 ytD <Ž.eFŽ.<55FM␣ which provesŽ. II . Conversely suppose thatŽ. I and Ž II . hold. Put VsÄ4gG:M55␣ -1 and X WsÄ4fgG:<²:f, Then by the Banach Alaoglu theoremwx 11, 3.15, p. 66 , W is weakU 2 yt n D compact. Let ކ s ÄeF4Ä4where tn is any sequence tending to 1 y as n tends to ϱ. ThenŽ. II shows that ކ ; W. Let us consider two cases: Case 1. There are infinitely many n say njj Ž.1, 2, . . . such that 2 s ys j D X eFŽ.where we denote tnjby s as a constant in G . In this case take j 2 X this constant as f and we have a sequence eys j D F converging to f in G . Case 2. ކ is an infinite set. Being an infinite subset of a compact set in the weakU topology of GX, ކ has a limit point in the weakU topology of GX, say f, by the Bolzano Weierstrass property. That is, every weakU neighbor- U hood of f contains an element of ކ. Fix g G. Then a typical weak X neighborhood of f is of the form NU Ž. s Äg g G : gŽ. g U4where Uis any neighborhood of the complex number fŽ.Ž seewx 11. . For j s 1, 2, . . . let Uj Äw : Using our standard notation used in Lemma 3.6 and putting sjjs 1 y ␦ Ž.so that ␦j tends to 0 q as j tends to ϱ we get from Lemma 3.7, uŽ.Ž.,tq␦jjs²:uz,␦ ,kŽ.yz,t .18 Ž. Allowing j ª ϱ inŽ. 18 we obtain uŽ.,ts²:fz Ž.Ž,kyz,t ..19Ž. Let us now observe thatŽ. 19 is valid for all t inŽ. 0, 1 despite the fact that for different t’s inŽ. 0, 1 the associated sequences Ä4␦j are also corresponding different. It is easy to check that for each g ޒ, kŽ. y z, t tends to k Ž. y z,1 in G as t ª 1 y Žit is easier to check that the Fourier transform of kŽ.yz,ttends to that of k Ž. y z,1 in G which will imply the required result by the continuity of inverse of the Fourier transform. . Hence allowing t ª 1 y inŽ. 19 and using Lemma 3.8 we get FwŽ.s²:fz Ž.,kw Žyz,1 .sfw˜ Ž.. X We remark that if for some g g G FwŽ.s˜gw Ž.then by Corollary 2.7, X X f s g in G proving the uniqueness of f as an element of G , in Theo- rem 3.9. 4. A COMPARATIVE STUDY In this section we prove the denseness of G in L Ž.␣,  for ␣ ) 0 and -0 thereby proving that L XŽ.␣,  is a sub-space of GX. Thus the Weierstrass transform theory will be directly applicable to L XŽ.␣,  in contrast with the classical situation. In the following we shall freely make use of the notations and results from Zemanianwx 12 . THEOREM 4.1. G is dense in L Ž.␣,  for ␣ ) 0 and  - 0. Proof. By definition L Ž.␣,  s D La, b where a ) ␣ ) 0 and b -  - 0. It is easily verified that G ; La, b for all a ) ␣ ) 0 and b -  - 0 and that L Ž.␣,  ; S, the space of rapidly decreasing functions. We shall use the sequence ÄŽ.n 4provided to us by Kenneth B. Howell in a ) 2 2 2 2 yz ynzr2 private communication where Ž.z s e , nŽ.z s ne for n g ގ z2 z2 2 and ) 0 with Ž.z s ey and Ž.z s ey r. Let us first show that if gLŽ.␣,, then ª in L Ž.␣,  as ª 0. Let g Lc, d for some c and d such that ␣ - a - c and d - b - . Then g La, b and we prove that ª in La, b as ª 0. Let us show that, for simplicity, for the GENERALIZED WEIERSTRASS TRANSFORM 525 semi-norm ␥11, ␥Ž.yª0as ª0. The cases for other semi-norms ␥k ,ks2, 3 . . . , follow similarly. By definition d ␥1Žy .ssup xta,b Ž.Ž. Ž.ty1 Ž.t tgޒ dt XX ssup xta,bŽ. Ž.t Ž.ty Ž.0q Ž.t Žt . tgޒ XX ssup xta,bŽ. Ž.tt Žt0 . tgޒ X qsup xta,bŽ. Ž.t Žt . tgޒ for some t0 between 0 and t by mean value theorem. Let us observe that X X Ž.zgGand hence < Ž.x Ž.k Žk. sup xtc,dnŽ.Ž) . Ž.ty Ž.t tgޒ Ž.k Žk. ssup xtc,dnŽ.ž/) Ž . Ž.ty Ž.t tgޒ n 22 Ž.k Žk. ynŽty.r2 s sup Hxtc,dŽ.Ž . Ž .y Ž.te d '2tgޒ ޒ n n22Žt.2 using the fact that Hedyyrs1 ž/'2ޒ n 22 Ž.kq1 ynŽty.r2 F sup Hxttc,dŽ.< Ž.for some t0in the open line segment joining t and n22 ␥ y1<<ynŽty.r2 F kq1Ž.sup Hxtxttc,dc Ž.,d Ž.0yed. '2 tgޒޒ Now we claim that for t0 in the open line segment joining t and where t, g ޒ, y1 B In fact keeping in mind that t0 lies in the open line segment joining t and , and that c ) 0 ) d ctyct0 dtydt0 < d< ctydt0 ct ctyc In the third combination we have assumed that F 0 since ) 0 and t G 0 together will imply that t0 G 0 which reduces to the first combina- tion. Similarly in the fourth combination we have assumed that ) 0. B n 22 ␥ ␥ << B In the above calculations the equality is obtained by a change of variable. Hence g Lc, dn« ) ª in L c,dn« ) ª in L a,b and ) ncg L ,d« Ž.)nnaª) in L ,b. Hence Ž.)nª in La, b as ª 0 and n ª ϱ. One can easily verify that g L Ž.␣,  implies Ž.)n gG᭙,nand we have shown that Ž.)n ªin LŽ.␣,. This proves the theorem. Now Theorem 1.9.1 ofwx 12, p. 24 implies the following corollary: X X COROLLARY 4.2. L Ž.␣,  is a sub-space of G . ACKNOWLEDGMENT The authors thank the referee for his valuable comments towards the improvement of the paper. GENERALIZED WEIERSTRASS TRANSFORM 527 REFERENCES 1. L. V. Ahlfors, ‘‘Complex Analysis,’’ McGraw᎐Hill, New York, 1979. 2. A. Brychkov and A. P. Prudnikov, ‘‘Integral Transforms of Generalized Functions,’’ Gordon & Breach, New York, 1989. 3. I. I. Hirshman and D. V. Widder, ‘‘The Convolution Transform,’’ Princeton Univ. Press, Princeton, NJ, 1955. 4. K. B. Howell, A new theory for Fourier analysis, Part 1, J. Math. Anal. Appl. 168 Ž.1992 , 342᎐350. 5. K. B. Howell, A new theory for Fourier analysis, Part 2, J. Math. Anal. Appl. 173 Ž.1993 , 419᎐429. 6. K. B. Howell, A new theory for Fourier analysis, Part 3, J. Math. Anal. Appl. 175 Ž.1993 , 257᎐267. 7. K. B. Howell, A new theory for Fourier analysis, Part 4, J. Math. Anal. Appl. 180 Ž.1993 , 79᎐92. 8. K. B. Howell, A new theory for Fourier analysis, Part 5, J. Math. Anal. Appl. 187 Ž.1994 , 567᎐582. 9. V. Karunakaran and N. V. Kalpakam, Boehmians and Fourier transforms, preprint, 1996. 10. V. Karunakaran and T. Venugopal, Convolution transform for generalized functions, Integral Transforms Special Funct., to appear. 11. W. Rudin, ‘‘Functional Analysis,’’ Tata McGraw᎐Hill, New Delhi, 1974. 12. A. H. Zemanian, ‘‘Generalized Integral Transformations,’’ Wiley᎐Interscience, New York, 1968.