The Bijectivity of the Tight Frame Operators in Lebesgue Spaces

Transcription

1 Iteratoal Joural o Aled Physcs ad atheatcs Vol 4 No Jauary 04 The Bectvty o the Tght Frae Oerators Lebesgue Saces Ka-heg Wag h-i Yag ad Kue-Fag hag Abstract The otvato o ths dssertato aly s whch ae rae wavelet systes sa Lebesgue saces have bee vestgated less The techue we used s aalogous to techue o alderó-zygud oerators but we rely o alderó-zygud decoosto theore We rove our a results wthout soothess assuto o rae wavelets We rove that ae tght rae wavelets sa Lebesgue saces uder the codto we also show that the ae tght rae oerator exteds ro L ( ) to a bouded lear ad bectve oerator o L ( ) or Uder such codto the ae orthooral bass o or L ( ) L ( ) s also a ucodtoal bass Idex Ters Bectve raes orthogoal bass ucodtoal bass wavelets I INTRODUTION The codtos o ay robles orce the use o a set o bass o a Hlbert sace H whch ca ot be orthooral (ote because o ucertaty rcles) Lastly we ca dsregard eve lear deedece we ed u wth the "raes" whose theory was rst develoed by Du ad Schaeer [] A rae acts le a overcolete set lear algebra where oe ca exress ay H wth but ot ecessarly a uue way sce the eleets eed ot to be learly deedet I ths aer we ocus o tght raes whch ther caocal dual rae has the sae structure as the raes theselves Ths les ay advatages Ideed t s ot oly dcult to cotrol the behavor o the dual raes but also othg ca guaratee the decay o the dual raes Tght raes act le orthogoal bases but are wthout lear deedece Hstorcally [] [3] [4] use the alderó-zygud oerators; [5] [6] use ultresoluto aalyss ad arcewcz terolato theore We rove that the rae oerator assocated the ae tght rae wth the codto exteds ro L ( ) to a bouded lear ad bectve oerator o L ( ) or By the way we ca use the sae techue to the case or a orthooral wavelet case We lst our a results Theore : The rae oerator S assocated (dee later) s bectve o L ( ) ad hece s dese the Lebesgue saces oreover S D or all L ( ) ad soe costats 0 D Theore : Let be a orthooral bass o L ( ) The ) T s o tye wea () ad o tye ( ) or all oreover T where does ot deed o or all L ( ) ) ors a ucodtoal bass or L ( ) or all 3) I s Bessela or L ( ) ad there exsts c 0 such that c { a } a { a } or all scalars a 4) I s Hlberta or L ( ) ad there exsts c 0 such that c { a } a { a } or all scalars a 5) s sultaeously Bessela ad Hlberta ad oly = auscrt receved Noveber 8 03; revsed Jauary 4 04 Ka-heg Wag s wth PhD Progra echacal ad Aeroautcal Egeerg Feg-ha Uversty 4074 Ta-hug Tawa (e-al: gtotoy98@galco) h-i Yag ad Kue-Fag hag are wth Deartet o Aled atheatcs Feg-ha Uversty 4074 Ta-hug Tawa (e-al: yesheslhao@galco chag@athcuedutw) II PRELIINARIES AND NOTATIONS Bases ths aer are Schauder bases I [7] t etos a secal class whch satses Weer codto Let be a easurable ucto o ad satses the ollowg eualty: DOI: 07763/IJAP04V453 46

2 Iteratoal Joural o Aled Physcs ad atheatcs Vol 4 No Jauary 04 W ( L l ) : su ( x ) [0) where s the characterstc ucto We cosder that ( x) dx ( x ) dx 0 su ( x ) W ( L l ) [0) It ollows that () holds we have L ( ) or all Let T be a ag ro ( ) or all The T s o tye ( ) () L to L ( ) T ( ) A L ( ) where A does ot deed o Slarly T s o wea tye ( ) A { x : T ( x) } 0 where A does ot deed o or ad s the Lebesgue easure We ote that a oerator o tye ( ) s o wea tye ( ) A seuece o eleets H s a rae or H there exsts costats A B>0 such that A B or all H Let be a rae wth rae oerator S (see [8]) The S or all H The seres coverges ucodtoally H { S } s called the dual o We also lst soe roertes whch are eeded to our results ) Let H L ( ) We deote the rae oerator S o as S : or all L ( ) T s called the re-rae oerator or the sythess oerator whch s bouded : ( ) { } : T l H T c c S TT T T ) Both S ad S - are o tye ( ) bouded vertble sel-adot ad ostve [8] 3) A rae ay have several duals; a dual whch s ot the caocal dual s called a alterate dual { S } ay ot have wavelet structure 4) { S } s also a rae H ad ts rae oerator s S - 5) For a tght rae{ } { S } { } ad A s A the tght rae boud There are rch orato about raes ca be oud [] III PROOFS We start at the L boudedess o rae oerators Let { : } be a rae L ( ) where We say satses / : ( ) 0 log ( x)su ( y) dx We lst two thgs that x y eed to be otced ) I s also belog to W L ( l ) so that s belog to L ( ) ) Our result ca be aled to all coactly suort orthooral/tght rae wavelets For all L ( ) by alderó-zygud decoosto theore [9] there exsts a collecto such that tervals { I } are dsot ( ) I : [ ( )) ( x) alost everywhere o F: \ I or all 0 ad also or all ( ) ( ) So we have I () ( ) We deote a ae tght rae : { : } ad P s the roecto ro L ( ) oto : sa{ : } ad let o or all { S : } be the caocal dual rae P ( ) : S A L ( ) For all g : P ( ) F I ( ) ( ) L L ( ) we set [ ] h : g P ( ) It ollows that Sh ( ) I I I las that S s o wea tye ( ) ad o tye ( ) whch reles o the sae dea o [5] [6] but we dot have colcated estatos 47

3 Iteratoal Joural o Aled Physcs ad atheatcs Vol 4 No Jauary 04 Theore 3: Uder otatos as above the S s wea tye ( ) ad o tye ( ) or all Fally we rove Theore by deg a oerator o L ( ) as T : or all L ( ) sce the eature o T s slar to the rae oerator S So we ca coclude that T s o tye ( ) or all Proo o Theore 3: Frst we cla that or all >0 g A where costat A does ot deed o ad Assug by Drchlet test : A su ( x ) ( ) ( ) ( ) 4 4 ( ) ( ) ( ) I I P ( ) I P ( ) P ( ) I I I I su ( x ) ( ) su ( x ) I 4 ( ) I su ( x ) 6 I N where by Drchlet test : su ( ) I N x ollows ro the ad Ideed t ( ) I I We also have uder such I I It leads that or each we have ( ) 4 Whe we tae ay te ters o ad Next I su ( x ) W ( L l ) F to F g 6 N Secodly we set t leads I : [ ( ) ( )) F : \ I ( ) x ( x) : su ( y) x 0 x y Sh F I Sh su ( x) I ( ) I su ( x ) I B Ideed xg we cosder I / / su ( ) I ad breag to arts x su ( x) [ ) ( ) ( ) 4 ( ) 48

4 Iteratoal Joural o Aled Physcs ad atheatcs Vol 4 No Jauary 04 4 (0) log ( ) 4 (0) (log x ) ( x) dx [ ) su ( x) [ ) su ( x) 4 ( ) 0 ( ) 4 (0) (log x ) ( x) dx Ths arguet also leads to su ( x) su ( x) I I su ( x) The suato W ( L l ) x I su ( x ) also te whch t s ot oly deedet wth x but there exsts a costat D su ( x ) D su ( x) x I Fally or all { F } 3/ L L ( ) 0 we ote that (ro ()) ad S s o tye ( ) { x : S } { x : Sg / } { x : Sh / } { x : Sg / 4} (B 3) / [ ] (4A B 3) / So S s o wea tye ( ) By arcewcz terolato theore t leads to that S s o tye ( ) hece S s o wea tye ( ) or all << ad thus by dualty or all Proo o Theore : Fro Theore 3 or all L L ( ) we have h hl ( ) su h hl ( ) h su S S h L ( ) or soe costat 0 ad / / Sce S L L ( ) s dese L ( ) we tae { } L L ( ) such that coverges to L ( ) We deote L or or all l l S S (3) S s the adot o S whch s o L ( ) to L ( ) Followg (3) S su ( S ) g g S g S g su ( ) g Ths leads to g S g Thus we ca coclude that S s bectve o L ( ) Proo o Theore : We rove (3) (4) It s a well-ow act whch every bouded ucodtoal bass s Bessela (Hlberta) L ([0]) ( ) [0] We dee I : [ ) J : [ ) J : [ ) ad there exsts a lear soetrc ag roe L ( J ) oto L ( I ) I : [0] J : Deg a a ro L ( ) oto L ([0 ]) by ( ) : ( ) so that s lear or reservg sce J ( ) ( ) L ([0]) J L ( I ) ( ) J ( ) L ( ) J L I L ( ) or all L ( ) Thereore { ( )} s also a bouded ucodtoal bass L ([0]) REFERENES [] R J Du ad A Schaeer A class o oharoc Fourer seres Tras Aer ath Soc vol [] K Ahad R Kuar ad L Debath Exstece o ucodtoal wavelet acet bases or the saces L ( ) ad H ( ) Tawaese J ath vol 0 o [3] E Herádez ad G Wess A Frst ourse o Wavelets Boca Rato Florda: R Press 996 [4] Y eyer Wavelets ad Oerators abrdge: abrdge Uversty Press 99 [5] G Greberg Wavelet bases L ( ) Studa ath vol 06 o

5 Iteratoal Joural o Aled Physcs ad atheatcs Vol 4 No Jauary 04 [6] Wotaszczy Wavelets as ucodtoal bases L ( ) J Fourer Aal Al vol 5 o [7] N Weer Taubera theores A o ath vol o [8] O hrstese A Itroducto to Fraes ad Resz Bases Brhauser Bosto 003 [9] E Ste Sgutarltegrals ad Deretablty Proertes o Fuctos Prceto NJ: Prceto Uversty Press 970 [0] I Sger Bases Baach Saces I ad II New Yor: Srger-Verlag 970 K Wag receved hs BS ad S degrees atheatcs ro Natoal heg Kug Uversty Feg ha Uversty 000 ad 003 resectvely He s a researcher PhD Progra echacal ad Aeroautcal Egeerg Sce 00 he has bee a lecturer the Deartet o Aled atheatcs at Feg ha Uversty Tawa Hs research terests clude rae theory ad wavelet aalyss I Yag receved her BS ad S degrees atheatcs ro Natoal Natoal Tachug Uversty o Educato Feg ha Uversty 999 ad 0 resectvely Sce 999 she has bee a teacher the Dogxg eleetary school Tachug Tawa Her research terests clude Neural Networs ad wavelet aalyss K F hag receved hs BS S ad PhD degrees atheatcs ro Natoal Tawa Noral Uversty Natoal Tsg Hua Uversty ad the Uversty o Texas at Aust ad 993 resectvely Sce 998 he has bee a Proessor the Deartet o Aled atheatcs at Feg ha Uversty Tawa Hs research terests clude aroxato theory ad wavelet aalyss 50