Generalized Wavelet Transform Associated with Legendre Polynomials

Transcription

1 Geeralized Waelet Trasor Associated with Legedre oloials C ade Aja Kuar Garg Egieerig College Ghaziabad Idia MM Dixit North Easter Regioal Istitute o Sciece ad Techolog Nirjuli-799 Idia Rajesh Kuar Noida Istitute o Egieerig ad Techolog Greater Noida Idia ABSTRACT The coolutio structure or the Legedre trasor deeloped b Gegebauer is exploited to deie Legedre traslatio b eas o which a ew waelet ad waelet trasor iolig Legedre oloials is deied A geeral recostructio orula is deried MSC 33A4; 4C Kewords Legedre uctio Legedre trasors Legedre coolutio Waelet trasors INTRODUCTION Special uctios pla a iportat role i the costructio o waelets atha ad Dixit [5] hae costructed Bessel waelets usig Bessel uctios But the aboe costructio o waelets is o sei-iiite iteral Waelets o iite iterals iolig solutio o certai Stur-Liouille sste hae bee studied b U Depczsi [] I this paper we describe a ew costructio o waelet o the bouded iteral - R usig Legedre uctio We ollow the otatio ad teriolog used i [7] Let deote the space L p p or C[-] edowed with the ors p / p p x dx C p sup x x A ier product o is gie b g x gx dx 3 As usual we deote the Legedre poloial o degree N b x ie x d! x ; x [-] dx For these poloials oe has i x ; x [-] 4 " ' ii x x x x x ; 5 iii ' 6 The Legedre trasor o a uctio is deied b L[ ] x x dx; The operator L associates to each coplex ubers Legedre coeiciets The ierse Legedre trasor is gie b L[ ] x x Lea Assue g i L[ ] ; 7 sequece o real called the Fourier x N ad c R the 8 ii L[ g] L[] L[g] iii i L[c] cl[]; L[ ] or all N i x ae ; L[ ] j j j j N Let us recall the uctio Kxz which plas role i our iestigatio Kx z x z xz z z z 9 otherwise z x [-x - ] / ad z x + [-x - ] / 35

2 The the uctio Kxz possesses the ollowig properties; i ii Kxz is setric i all the three ariables K x zdz Also it has bee show i [8] that x zkx z dz Applig 8 to we hae K x z x z The geeralized Legedre traslatio or [ ] o a uctio is deied b x x z Kx zdz Usig Hölder s iequalit it ca be show that ad the ap ito itsel 3 is a positie liear operator ro As i [7] or uctios g deied o [-] thegeeralized Legedre coolutio is gie b *g x x g d z g Kx zd dz 4 Lea I gl the the coolutio *g x exists ae ad belogs to Moreoer *g 5 *g g l g 6 For a L the ollowig arseal idetit holds or Legedre trasor 7 I this paper otiated ro the wor o classical waelet trasors c [] [3] we deie the geeralized waelet trasor ad stud its properties A geeral recostructio orula is deried A recostructio orula uder a suitable stabilit coditio is obtaied Furtherore discrete LWT is iestigated Usig Legedre Waelet rae ad Riesz basis are also studied A ew exaples o LWT are gie Siilar costructios o waelets ad waelet trasors o seiiiite iteral ca be oud i [4] ad [5] GENARALAISED WAVELET TRANSFORM For a uctio deie the dilatio Da b D a t at a Usig the Legedre traslatio ad the aboe dilatio the waelet ba ba t is deied as ollows: t bdat bat Kb tz azdz 3 b ad a The itegral is coerget b irtue o 3 Now usig the waelet ba the Legedre waelet trasor LWT is deied as ollows: a L ba t b t t ba t t az dt Kb tzdz proided the itegral is coerget b dt a Sice b 3 ad wheeer b Lea the itegral 6 is coerget or L The adissibilit coditio or the Legedre waelet is gie b A 7 Fro 7 it ollows that But Yields t t dt t tdt tdt Hece t chages sig i - thereore it represets a waelet - 36

3 deies a Legedre waelet ad Theore I L the the coolutio deies a Legedre waelet L roo Let ad so that is a bouded uctio o - B Lea We hae Thereore L so that a a L ba bdb ; a A Thereore represets a Legedre waelet Theore Let L ad adl ba be the cotiuous Legedre waelet trasor The we hae the ollowig iequalit L ba l roo The aboe iequalit ollows ro 5 3 A GENERAL RECONSTRUCTION FORMULA I this sectio we derie a geeral recostructio orula ad show that the uctio ca be recoered ro its Legedre waelet trasor Usig represetatio 6 we hae L ba t az Kb tz dzdt t az dz dt b t z b t tdt az z dz a az zdz b a 3 a Multiplig both sides b ad a weight uctio qa ad itegratig both sides with respect to a ro to we hae qa a L ba bdb da qa a da 3 Assue that Q qa a da Usig 33 3 ca be writte as Q Q We hae ro 3 ba 33 qa a L ba bdbda qa L ba a bdbda t az 34 Kbtz az dz a b t b t z dz b t az - b t a 35 z dz Usig 35 i 34 we hae Q Set ba qa b a /Q L ba 36 ba dadb 37

4 The ba q a L ba dadb 37 uttig 37 i 8 we hae t t qa L ba Thereore t 38 ba qal ba qa L ba ba ba tdadb 4 THE DISCRETE TRANSFORM The cotiuous Legedre waelet trasor o the uctio i ters o two cotiuous paraeters a ad b ca be coerted ito a sei-discrete Legedre waelet trasor b assuig that a -; Z ad b L Now we assue that satisies the so called stabilit coditio A B 4 or certai positie costats A ad B < A B L The uctio satisig 4 is called dadic waelet Usig the deiitio 4 we deie the sei-discrete L Legedre waelet trasor o a b b L b t t L b t b z t dt z Z 4 Now usig arseal idetit 7 4 ields the ollowig A L B L 45 Theore 4 Assue that the sei-discrete LWT o a L is deied b 5 Let us cosider aother waelet * deied b eas o its Legedre trasor dadb tdadb The t * L j * b j t 46 b db 47 L roo: I iew o 44 or a we hae L L b b * * t * t L t * t t b db t b db b bdb t L * t j The aboe theore leads to the ollowig deiitio o dadic dual L Deiitio 4 A uctio is called a L dadic dual o a dadic waelet i eer ca be expressed as t L b ~ t bdb 48 So ar we hae cosidered sei-discrete Legedre waelet L trasor o a discretizig ol ariable ~ j 38

5 a Now we discretize the traslatio paraeter b also b restrictig it to the discrete set o poits b ; b b [ ] b Z N 49 is a ixed costat We write t b ;a t t b 4 The the discrete Legedre waelet trasor o a L ca be expressed as L b N4 a ; b ; Z The stabilit coditio or this recostructio taes the or 4 A b ; L B Z N A ad B are positie costats such that A B Theore 43 Assue that the discrete LWT o a L is deied b 4 ad stabilit coditio 4 holds Let T be a liear operator o L- deied b The T Z N T ; b b ; b ; b ; b ; b Z N roo Fro the stabilit coditio 4 it ollows that the operator deied b 43 is a oe-oe bouded liear operator Set g T The we hae T Thereore A T g T g - L Z N A g g T b ; g T so that T g A g L Hece eer ca be recostructed ro its discrete LWT gie b 4 Thus T T Fiall set Z N T ; b b ; b ; T b ; Z N 45 The the recostructio 45 ca be expressed as Z N b ; b which copletes the proo o theore 43 5 FRAMES AND RIESZ BASIS IN L- I this sectio usig basis L- is studied b ; a rae is deied ad Riesz o L Deiitio 5 A uctio is said to o L b ; geerate a rae with saplig rate b i 5 holds or soe positie costats A ad B I A B the the rae is called a tight rae L Deiitio 5 A uctio is said to b ; geerate a Riesz basis o with saplig rate b i the ollowig two properties are satisied The liear spa b 5 ; : Z > is dese i L- There exist positie costats A ad B with <A such that A { c B } c bo; B { c } Z N 5 { c } or all N Here A ad B are called the { b ; } Riesz bouds o L Theore 53 Let the the ollowig stateets are equialet 39

6 { b ; } { b ; } is a Riesz basis o L-; is a rae o L- ad is also a - liearl idepedet ail i the sese that i b ; c {c } ad the c Furtherore the Riesz bouds ad rae bouds agree roo It ollows ro 5 that a Riesz basis is liearl idepedet { b ; } - Let be a Riesz basis with Riesz bouds A ad B ad cosider the atrix operator M rs rs N N the etries are deied b rs b ;rs b ; 53 The ro 5 we hae A {c } crs rsc B {c } rs so that M is positie deiite We deote the ierse o M b M which eas that both tu ad B rs;tu { c rs rs N tu; } r s; 54 rs - c rs c A { } r s c r s are satisied This allows us to itroduce rs x rs; b ; x N rs L Clearl ad it ollows ro 53 ad 55 that rs ; ; b ; r s r s which eas that {rs} is the basis o L- which is { } b ; dual to N Furtherore ro 65 ad 66; we coclude that rs rs ad the Riesz bouds o {rs} are B- ad A- L I particular or a we a write B ad x bo; bo ; x A - bo; 58 Sice 58 is equialet to 4 thereore stateet i iplies stateet ii To proe the coerse part we recall g L Theore 43 ad we hae or a ad T-g gx Z N bo; bo; Also b the liear idepedece o represetatio is uique Fro the Baach-Steihaus ad ope { b ; } appig theore it ollows that is a Riesz basis o L- { } b ; this 6 REFERENCES [] CK Chui A Itrodcutio to Waelets Acadic ress New Yor 99 [] U Depczsi Stur-Liouille waelets Applied ad Coputatioal Haroic Aalsis [3] G Kaiser A Friedl Guide to Waelets Birhauser Verlag Bosto 994 [4] RS atha Fourier-Jacobi waelet trasor Vijaa arishad Aushadha atria [5] RS atha ad MM Dixit Cotiuous ad discrete Bessel Waelet trasors J Coputatioal ad Applied Matheatics [6] ED Raiille Special Fuctios Macilla Co New Yor 963 [7] RL Stes ad M Wehres Legedre Trasor Methods ad Best Algebraic Approxiatio Coet Math race Mat IJCA TM : wwwijcaolieorg 4