A q 2 -Analogue Operator for q 2 -Analogue Fourier Analysis

Transcription

1 JOURNAL OF MATEMATICAL ANALYSIS AND APPLICATIONS, ARTICLE NO AY A -Aalogue Operator for -Aalogue Fourier Aalysis Richard L Rubi Departmet of Mathematics, Florida Iteratioal Uiersity, Miami, Florida 3399 Submitted by William F Ames Received Jauary 3, 997 INTRODUCTION I this paper we itroduce -aalogue differetial operators adapted to study certai -aalogue fuctios ivestigated by T Koorwider ad R F Swarttouw We shall cosider the -aalogue trigoometric fuctios itroduced i 0, ad cos z; 0; ;, z z Ý ; 0 ; zj z; 3 ; z 3 si z; 0; ;, z 4 z Ý 5 ; 0 ; zj z;, 6 ; X97 $500 Copyright 997 by Academic Press All rights of reproductio i ay form reserved

2 57 RICARD L RUBIN usig the -aalogue Bessel fuctio itroduced by Exto, ad W ah 5, ; J z; z 0; ;, z 7 ; ere we are usig the basic otatioal covetios of 0, 4 I particular, let 0, defie a; Ł a j, Z, a; j0 0, a; lim a;, ad z 0; b;, z Ý 8 b; ; 0 0; b; ; z defies a etire aalytic fuctio i z provided b is outside the set,,, 4 A ey property of J z;, cosz;, ad siz; is that they satisfy appropriate -aalogue orthogoality relatios cf 0 I fact, the - aalogue Bessel fuctios ad closely related variats have received much attetio because of their importace i the study of -aalogues of represetatios of the Group of Plae Motios ad of the Quatum Group of Plae Motios, -differetial euatios, ad other topics additio to the wor already cited, see, eg, Vasma ad Korogodsii I, Kalis, Miller, ad Muherjee 6, Koeli ad Swarttouw 8, Koeli 7, ad Swarttouw ad Meijer owever, the fact that cosz; ad siz; have disjoit sets of eigevalues with respect to the classical -differetial operator f z f z Df z 9 z ad also with respect to D D has limited their cosideratio ad discouraged efforts to costruct a -expoetial built from fuctios defied by cos z; ad siz; I this paper we cosider the operator f z f z f z f z f z f z, 0 z ad its reormalized versio

3 -ANALOGUE OPERATOR 573 We will show that or is a useful -aalogue of the derivative i that it produces aalogues of the stadard differetial relatioships be- twee cos z; ad si z; We will defie a -aalogue expoetial fuctio i terms of these fuctios, study some of its properties, ad use it to defie ad study a -aalogue Fourier Trasform To coclude the itroductio, we would lie to discuss some differeces betwee ad D istorically, the relatio betwee -aalogues ad the classical hypergeometric fuctios is based o observatios such as lim so that, lim z ; z! Also see Remar later i this paper I our cotext, we have the limits lim cos z; cos z lim si z; si z 3 Aother classical fact is that if f is differetiable at z, lim D f z f z 4 We also have for f differetiable at z, lim f z f z 5 owever, whe studyig expressios such as ad 3, freuetly we are iterested i limits of the form lim T f z; where both the operator ad its argumet fuctio chage with Computig the classical -derivative of the fuctios i ad 3 gives D si z; cos z: 6 D cos z; si z; 7 Although, lim D si z; cos z ad lim D cos z; si z, we see that the eigevalue relatioship betwee these fuctios ad D is ot aalogous to the classical situatio for 0 O the other had, si z; cos z; 8 cos z; si z; 9

4 574 RICARD L RUBIN givig the classical limit as ad, more importatly, a useful aalogue relatioship for 0 These relatioships will allow us to defie a -aalogue expoetial which exhibits appropriate behavior uder I additio to studyig the iteractio of ad related operators with the -aalogue trigoometric fuctios ad with a -aalogue expoetial fuctio ad applicatios, the operator ca also be used for other aalogue results For example, we have a umber of -formulas for -Bessel fuctios of iteger order which yield classical Bessel fuctio idetities i the appropriate limit Correspodig formulas i terms of D have bee obtaied by others, cf 0, 8, ad refereces foud i these papers TE OPERATOR We use the otatio f z f w z w where w represets a complex costat Uless otherwise specified we will always assume that fuctios f are defied o sets, S, which are symmetric i the sese that if z S the z S ad z S I this sectio will always be iteger-valued Our first lemma lists some useful computatioal properties of ad, ad reflects the sesitivity of these operators to the parity of their argumets LEMMA a If f is a odd fuctio, f z D f z a If f is a odd fuctio, f z z f z f z b If f is a ee fuctio, f z D f z b If f is a ee fuctio, f z z f z f z c If f ad g are both ee, fg z f z g z g z f z d If f ad g are both odd, fg z f z g z g z f z e For ay complex umber, f z f z The verificatio of these formulas is straightforward ad will be left to the reader We cosider a cotext i which the behavior of is very useful Usig the -aalogue trigoometric fuctios defied i ad 5 we defie a -aalogue expoetial by e z; cos iz; i si iz; 0 z z Ý Ý ; ; 0 0

5 -ANALOGUE OPERATOR 575 e z; is absolutely coverget for all z i the plae, 0, sice both of its compoet fuctios are lim e z; e z poit- wise ad uiformly o compacta, because both of its compoet fuctios satisfy correspodig limits by the followig remar u Remar u lim 0; ;, z F 0 ; u ; z, uiformly i z i compact subsets of the plae ere F deotes the geeralized 0 hypergeometric series The remar is proved i 0, p 459 The basic computatioal formulas for these -aalogue fuctios are give by: LEMMA a si z; cos z; b cos z; si z; 3 c e z; e z; 4 Proof cosz; is a eve fuctio, so z cos z; Ý 0 ; ž z / z Ý ; z Ý ; m mm m z Ý ; m m0 si z; Part a is proved similarly, ad c follows from a ad b usig the liearity of ad Lemma e 3 A -ANALOGUE FOURIER TRANSFORM We will use the otatio for -itegrals itroduced by Jacso, cf 4, f t dt Ý f 0

6 576 RICARD L RUBIN ad Set f t dt Ý f f 4 p p p p L d f : f p f t d t Ý f ½ for p, ad set 4 4 L d f : f sup f : Z p f 4 5 The followig properties ca be verified by direct calculatio LEMMA 3 If f t dt exists, a f odd implies f t d t0 b fee implies f t d t f 0 t dt s c s a iteger implies f t dtf t s dt We will also use the followig lemma Let F* t sgtf t LEMMA 4 If f * t g t d t exists, f * t g t dt f t g * tdt Proof The verificatio of this results is a tedious calculatio usig the defiitios of the -itegral give above ad of We will give a typical term: f g f g f g Ý Ý Ý f g Ý z We will eed the -Gamma fuctio defied by ; ; z, cf 4 Note that lim z z, see 9 Defie the -aalogue Fourier Trasform to be ˆf x; f t e i tx; d t 5

7 -ANALOGUE OPERATOR 577 m If we impose the coditio that be i 0, : for some iteger m4 ad if we let uder this side coditio, we obtai, formally, the classical Fourier Trasform o the lie See also the commet after the ext lemma Therefore, i the remaider of this paper, we assume that 0, : m for some iteger m 4 6 It should be oted that if we disregard the limit as, we ca formulate a defiitio of the Fourier Trasform which will satisfy all the correspodig versios of the results discussed below for all 0, For coveiece, set C 7 If we show that e i ; is bouded for all itegers, the it will follow immediately that the -Fourier Trasform defies a bouded liear operator from L d to L d We tur to this tas We use the followig results of Koorwider ad Swarttouw 0 : LEMMA 5 a For z ad, m itegers, z ; m Ý ; z ; m m z 0; z ;,, ; z 0; z ;, where the sum coerges absolutely, ad uiformly o compact subsets of the ope uit dis b For f L d, or 0 g C cos t ; f t d t implies 0 f C cos s ; g s d s 0 g C si t ; f t d t implies 0 f C si s ; g s d s

8 578 RICARD L RUBIN A study of the covergece as of the cosie trasform defied i part b of this lemma ca be foud i 3 We obtai some useful ieualities by combiig this result with the followig classical eualities of Rogers ad Ramauja, p 8, Ý Ł 0, 8 ; CI 4 where CI Z :, 0 mod0, mod0, 8 mod0 Ý Ł 0 ad, 9 ; SI 4 where SI Z :, 0 mod0, 3 mod0, 4 mod0 LEMMA 6 cos x;, si x:, ad e i x; are all bouded for x : is a iteger 4 I fact cos x; Ł ad CI si x; Ł SI Proof Lettig m ad replacig by i part a of Lemma 5 gives z ; ; Ý z 0; z ;, 30 Taig z, 0 i 30 shows that ; ; cos ; 0; ;, 3 3 for all itegers,0 Alterately, taig z, 0i 30 gives that 3 0; ;, ; 3 si ; 3 ; for all itegers, 0 This shows that x cosx; ad x si x; are bouded for x : is a iteger 4 Thus cos x; is bouded for x : is a o-positive iteger 4

9 -ANALOGUE OPERATOR 579 Now for z, usig the expasio 5 we get si z; Ý Ł 33 0 ; SI The last euality follows from 9 Moreover the fact that implies that ; Ł Ł ad ; ; Combiig these ieualities, ; Ł ; SI This, combied with 3 ad 33 gives si ; Ł for a iteger 34 SI A similar argumet usig 8 shows cos ; Ł, for a iteger 35 CI Fially, usig the eveess of cos x;, we see that it is also bouded with the same boud for x : is a iteger 4 Applyig coditio 6, we see that cos x; is bouded with the above boud for x : is a iteger 4 A similar argumet shows that si x; is bouded for x : is a iteger 4 Fially, combiig the above, we see that ei x; cos x; si x; is bouded o the same set At this poit, eve though we do ot have a simple additio formula for ez;, we ca establish several -aalogue Fourier trasform results usig essetially stadard argumets For example, it is easy to show: COROLLARY If f, g L d, the ˆ ˆ ˆ s s ˆ s ˆ a f is a bouded liear operator from L d to L d b f t; g t d t f t g t; d t c For s a iteger, f x f x d If uf u L d, f x; i ufu x

10 580 RICARD L RUBIN A complemet to part d of the above corollary is: COROLLARY If f * L d : f * x: i f* x; Proof The formula follows by expadig the left-had side usig the defiitio of the aalogue Fourier trasform ad the applyig Lemma 4 ad usig Lemmas e,, ad 6 We tur to the L theory of the -aalogue Fourier Trasform L L d is dese i L d Cosider fuctios with fiite support Sice the -aalogue Fourier Trasform is defied ad bouded o L L d for such fuctios, it defies a bouded extesio to all of L d We ca use Lemmas 3 ad 5 to prove a iversio theorem ˆ TEOREM f L L d implies f C f t; e i t; d t Proof We begi by rewritig the trasform pair of Lemma 3 b as 0 0 f C C cos ts; f t d t cos s ; d s f C C si ts; f t d t si s ; d s 37 As far as possible, we will apply a stadard strategy to derive Fourier Iversio Namely, write f fe fod with fe eve ad fod odd Usig 36 ad the Lemma 3 b ad a we get e e 0 0 f 4C cos t ; f s cos st; d sd t e 0 C cos t ; f s cos st; d sd t 0 C cos t ; f s cos st; d sd t Similarly, we get od 0 f C si t ; f s si st; d sd t

11 -ANALOGUE OPERATOR 58 Combiig these expressios ad usig the eveess of the itegrad i the t-variable gives f 0 C si t ; si st; 4 cos t cos st; f s d sd t C si t ; si st; 4 cos t cos st; f s d sd t Sice eix; cosx; i six;, we see that e i x; e i y; cos x; cos y; si x; si y; i si x; cos y; si y; cos x; Usig Lemma 6 ad the fact that f L d, apply Fubii s Theorem to show that si t ; cos st; si st; cos t ; d sd t ½ si t ; cos st; si st; cos t ; d t f s d s This last itegral is zero by Lemma 3 a sice the itegrad of the t-itegral is odd i t Applyig this to the above expressio for f yields ½ 5 f C C e i t ; e i st; f s d sd t C e i t ; C e i st; f s d s d t ˆ C e i t ; f t; dt 5

12 58 RICARD L RUBIN Now we ca establish a aalogue to the Placherel Theorem I what 4 ˆ follows assume x : is a iteger Defie f x; f x; Theorem says that f L L implies f x f ˆ x Also, if f z is the complex cojugate of f z, ote that e ix; e ix; Thus ˆ ˆ f x; Cf t e i tx; dt f x; For f L L d, say, with fiite support, usig Corollary b, f t f t dt ˆ ˆ ˆ f t f t d t f t f t; d t f t; f t; dt ˆ ˆ f t; f t; dt Sice the fuctios with fiite support are dese i L d, we get ˆ COROLLARY 3 f L d implies f f ; REFERENCES G Adrews, Series Their Developmet ad Applicatio to Aalysis, Number Theory, Combiatorics, Physics ad Computer Algebra, Amer Math Soc, Providece, 986 Exto, Basic ypergeometric Fuctios ad Applicatios, Ellis orwood, Chichester, M Fichtmuller ad W Weich, The limit trasitio of the -Fourier trasform J Math Phys , G Gasper ad M Rahma, Basic ypergeometric Series, Ecyclopedia Math Appl, Vol 35, Cambridge Uiv Press, Cambridge, W ah, Die mechaishce Deutug eier geometrische Differezegleichug, Z Agew Math Mech , E G Kalis, W Miller ad S Muherjee, Models of -algebra represetatios: The group of plae motios, SIAM J Math Aal 5 994, T Koeli, The uatum group of plae motios ad the ah-exto -Bessel fuctio, Due Math J 76, 994, T Koeli ad R F Swarttouw, O the zeroes of the ah-exto -Bessel fuctio ad associated -Lommel Polyomials, J Math Aal Appl , T Koorwider, Jacobi fuctios as limit cases of -ultraspherical polyomials, J Math Aal Appl , T Koorwider ad R F Swarttouw, O -aalogues of the Fourier ad ael trasforms, Tras Amer Math Soc , R F Swarttouw ad G Meijer, A -aalogue of the Wrosia ad a secod solutio of the ah-exto -Bessel differece euatio, Proc Amer Math Soc 0 994, L L Vasma ad L I Korogodsii, A algebra of bouded fuctios o the uatum group of the motios of the plae, ad -aalogues of Bessel fuctios, Soiet Math Dol , 7377