Asymptotic Expansions of Legendre Wavelet

Transcription

1 Asptotic Expasios of Legede Wavelet C.P. Pade M.M. Dixit * Depatet of Matheatics NERIST Nijuli Itaaga Idia. Depatet of Matheatics NERIST Nijuli Itaaga Idia. Astact A e costuctio of avelet o the ouded iteval - ) R usig Legede poloial ad the avelet expasio i tes of Legede poloial is peseted. Keods: Legede poloial; Legede Wavelet tasfo. INTRODUCTION Special fuctios pla a ipotat ole i the costuctio of avelets. Patha ad Dixit [] have costucted Bessel avelets usig Bessel fuctios. But the aove costuctio of avelets is o sei-ifiite iteval ). Wavelets o fiite itevals ivolvig solutio of cetai Stu-Liouville sste have ee studied U. Depczsi []. I this chapte e descie a e costuctio of avelet o the ouded iteval - ) R usig Legede poloial ad the avelet expasio i tes of Legede poloial is peseted. Let deote the space L p ) p o C[-] edoed ith the os p C f x) p dx / p p.) sup f x)..) x A ie poduct o is give f g f x) gx) dx.3) As usual e deote the Legede poloial of degee N P x) i.e. P x) d! x ) ; x [-]. dx Fo these poloials oe has i) P x) P ) ; x [-].4) " ' ii) x ) P x) x P x) ) P x) ;.5) iii) P ' ) )..6) The Legede tasfo of a fuctio f is defied L [f ]) f ) f x)p x)dx; N.7) The opeato L associates to each coplex) ues f ) coefficiets. The ivese Legede tasfo is give f sequece of eal called the Fouie Legede v L [f ] x) f x) ) f ) P x)..8) Lea.. Assue f g i) L[f ]) f ; N ad c R the ii) L[f g]) L[f] ) L[g] ) iii) iv) L[cf]) cl[f]); L [f ]) fo all N iff fx) = a.e ; L [P ] j) j j j) N Let us ecall the fuctio Kxz) hich plas ole i ou ivestigatio Kx z) x z xz otheise z z z hee z = x [-x ) - )] / ad z = x + [-x ) - )] /..9) The the fuctio Kxz) possesses the folloig popeties; i) Kxz) is setic i all the thee vaiales 887

2 ii) K x z)dz. Also it has ee sho i [3] that P x) P ) P z)kx z) dz.) Applig.8) to.) e have K x z) ) P x) P ) P z).) The geealized Legede taslatio fo [ ] of a fuctio f is defied f )x) f x ) f z) Kx z)dz.) Usig Hölde s iequalit it ca e sho that ad the ap ito itself..3) f f is a positive liea opeato fo As i [3] fo fuctios fg defied o [-] the geealized Legede covolutio is give f *g) x) f ) f ) x f z) x) g) d ) g) d g) Kx z) d dz.4) Lea.. If f gl ) the the covolutio f*g) x) exists a.e.) ad elogs to. Moeove f *g.5) f *g) The poof ca e foud i [3]. g l ) f )g)..6) Fo a f L ) the folloig Paseval idetit holds fo Legede tasfo ) f ) f..7) LEGENDRE WAVELET Fo a fuctio defie the dilatio D a D a t) at) a..) Usig the Legede taslatio ad the aove dilatio the avelet t) is defied as follos: a t) D t) at).) a a Ktz) az)dz.3) hee ad a. The itegal is coveget vitue of.3). No usig the avelet ψ a the Legede avelet tasfo LWT) is defied as follos: a L f ) a) ft) t)..4) f t) a t) dt.5) f t) az) Povided the itegal is coveget. Sice.3) ad.) K tz)dz dt heeve a.6) Lea. the itegal.6) is coveget fo f L ). The adissiilit coditio fo the Legede avelet is give ) A.7) Fo.7) it follos that ). But ) t) P t) dt ields ) t) P t)dt. t)dt Hece t) chages sig i -) theefoe it epesets a avelet. THE DISCRETE TRANSFORM The cotiuous Legede avelet tasfo of the fuctio f i tes of to cotiuous paaetes a ad ca e - 888

3 coveted ito a sei-discete Legede avelet tasfo assuig that a = - ; Z ad. No e assue that L ) satisfies the so called stailit coditio A ) B 3.) fo cetai positive costats A ad B < A B. The fuctio L ) satisfig 3.) is called dadic avelet. Usig the defiitio.4) e defie the sei-discete Legede avelet tasfo of a f L ) L f ) ) L f ) ) f t) hee t) 3.) z) f t) t) dt 3.3) f ) 3.4) z) Z. No usig Paseval idetit.7) 3.) ields the folloig: A Lf B f L ) ~. 3.5) Defiitio 3.. A fuctio L ) is called a dadic dual of a dadic avelet if eve f L ) expessed as L f ca e ) ~ )P t) )d. f t) 3.6) So fa e have cosideed sei-discete Legede avelet tasfo of a f L ) discetizig ol vaiale a. No e discete the taslatio paaete also estictig it to the discete set of poits Z N 3.7) hee [ ] is a fixed costat. We ite t) t) t ) 3.8) ; ;a The the discete Legede avelet tasfo of a f L ) ca e expessed as L f ) a ) f Z N. 3.9) ; ; The stailit coditio fo this ecostuctio taes the fo A f ; B f L ) Z N 3.) hee A ad B ae positive costats such that A B. FRAMES AND RIESZ BASIS IN L -). I this sectio usig ; a fae is defied ad Riesz asis of L -) is studied. Defiitio 4.. A fuctio L ) is said to geeate a fae of L ) ith saplig ate if ; 3.) holds fo soe positive costats A ad B. If A = B the the fae is called a tight fae. Defiitio 4.. A fuctio L ) is said to geeate a Riesz asis of ith saplig ate if the ; folloig to popeties ae satisfied. i) The liea spa : Z > is dese i L -) 4.) ; ii) Thee exist positive costats A ad B ith <A B such that A {c Fo all } c o; B {c } Z N { c } ouds of } { ; N ORDER OF APPROIMATION 4.) ). Hee A ad B ae called the Riesz I this sectio folloig the techique of Depczsi [] a discussio o the expasio of f L ) i avelet seies is give ad ode of avelet coefficiet is otaied. Let N N e a stictl iceasig sequece of atual ues ad deote V the space V spa f : i 5.) i 889

4 hee of Hilet space = { --- N } is a idex set ad {f i} is the asis L ). The spaces V ae liea ad closed suspaces of L ) ad V. Moeove V V L ). 5.) The othogoal copleet of V i V + is deoted W i.e. e have V V W 5.3) Fo 5.) ad 5.3) e have V W W L ). 5.4) No e stud the appoxiatio popeties of the spaces V. Let us coside the Legede diffeetial equatio " ' x ) P x) x P x) ) P x) 5.5) togethe ith the folloig hoogeeous ouda coditios U f ) af ) a f ) Ul f ) f ) f). The eigevalues of the aove ouda value pole ae N. give Let P P P --- e the coespodig eigefuctios. No e itoduce the space f D L[ ];U f ) ad U f ) hee L f C f L [ ] '[ ]; " ). We ill alas assue that zeo is o eigevalue of 5.5). Fo [.p 36] e o that the Gee's fuctio gx ) of 5.5) is give gx ) P x)p ) x ) [ ] [ ] ad G : L ) L ) defied G f ) gx ) f) ) d is a copact opeato ith age D. Fo N set G G o o G ties) The iteated spaces D D G f ) : f L ). N ae the defied Note that D L ) ad D D ad D D. Fo [ p. 37] e also ote that f D fp 5.6) We ecall the folloig esult fo [ p.37] aout appoxiatio ith the spaces V = spa {P P ----P N} N < N +. Theoe 5.. Let N P f f P P deote the othogoal pojectio of f oto V. The fo D f P f O N Let. N ad f 5.7) ; e the discete Legede avelets defied 3.8). The fo a f L ) thee exists a sequece {c } such that M Q 5.8) f c hee M = N + N. ; Suppose Q : L ) L ) e the opeato such that Q = P + P hee P is the pojectio opeato defied i Theoe 5.. The age Q = W hee V + = V W. Folloig Chui [4] e have the folloig defiitio fo the Riesz asis of the spaces W hich ill e used i sequel. The discete legede avelet ; fos a Riesz asis of the spaces W if thee exist costats A ad B ith A B such that A c c ; B c fo all sequeces {c } such that c. 5.9) 89

5 Theoe 5.. Let } e a Riesz asis of the spaces { ; W ith Riesz oud A >. The fo eve fd Poof. Q / c O N N ad. 5.) f P f Pf f Pf f P f Fo 5.7) it follos that Q f O N. 5.) Usig the Riesz stailit coditio 5.9) ad elatio 5.8) e have f c ; A c Q so that ; c Qf A. Fo 5.) it follos that / c ON. 5.) hich copletes the poof of Theoe. Theoe 5.3 Let ; e a Riesz asis of the spaces W ith uppe Riesz oud B<. Assue that N fo soe. If N ad f L ) ith c / the f D. O N Poof. Let us coside the patial su N N Fo f T of seies f T. N N = e otai N 4 N f T N f T. 5.3) No Q f = P + P f. Theefoe Q f P f Pf N N f T 5.4) Usig the Riesz stailit coditio 5.9) e have N f T Qf B c 5.5) N Next usig the assuptio c CN fo 5.3) 5.4) ad 5.5) it follos that N f T N C N C B N. N N N / 4-4- Let C > ith C 4 The fo the covegece of N B C fo all N. N N f T C N N it follos that N. Fo 5.6) it easil follos that f D. This coplete the poof. REFERENCES - [] R.S. Patha ad M.M. Dixit Cotiuous ad discete Bessel avelet tasfos J. Coputatioal ad Applied Matheatics 6 3) 4-5. [] U. Depczsi Stu-Liouville avelets Applied ad Coputatioal Haoic Aalsis 5 998) [3] R.L. Stes ad M. Wehes Legede tasfo ethods ad est algeaic appoxiatio Aeitseicht Lehstuhl A fü Matheati Rheiisch Westfalische Techische Hochschule Aache Gea 978). [4] C.K. Chui A Itoductio to Wavelets Acadeic Pess Ne Yo 99). 4 89