Generalized Self-Similarity

Transcription

1 Jounal of Mathematcal Analyss and Alcatons 230, Atcle ID jmaa , avalable onlne at htt: on Genealzed Self-Smlaty Calos A. Cabell*, and Usula M. Molte Deatamento de Matematca, Facultad de Cencas Exactas y Natuales, Unesdad de Buenos Aes, Cudad Unestaa, Pabellon I, 1428 Catal Fedeal, Agentna Submtted by Wllam F. Ames Receved July 20, 1998 We ove the exstence of L functons satsfyng a knd of self-smlaty condton. Ths s acheved by solvng a functonal equaton by means of the constucton of a contactve oeato on an aoate functonal sace. The soluton, a fxed ont of the oeato, can be obtaned by an teatve ocess, makng ths model vey sutable to use n alcatons such as factal mage and sgnal comesson. On the othe hand, ths genealzed self-smlaty equaton ncludes matx efnement equatons of the tye fž x. Ý c fž Ax k. k whch ae cental n the constucton of wavelets and multwavelets. The esults of ths ae wll theefoe yeld condtons fo the exstence of L -efnable functons n a vey geneal settng Academc Pess Key Wods: self-smlaty; functonal equaton; dlaton equaton; efnement equaton; wavelets; fxed onts; factals; nvese oblem fo factals. 1. INTRODUCTION Self-smla objects ae those that can be constucted out of smalle coes of themselves. When we deal wth sets, ths concet can be fomulated usng the noton of teated functon schemes Ž IFS. Ž24, 4. : If Ž X, d. s a metc sace and f w,...,w 4 Žw: X X, w 4. 1 n 1,..., N s a set of mas, then A X s self-smla wth esect to f A w Ž A..It can be shown, that f X s comlete, and the mas ae contactve, then thee exsts a unque comact self-smla set wth esect to. Ths concet can be extended n dffeent ways to dffeent knd of objects: self-smla measues can also be defned usng IFS Žsee 24, 4. Both authos ae membes of the CONICET and ae atally suoted by Gant UBACyT EX048 and a secal gant of the Fundacon Antochas. E-mal addess: ccabell@dm.uba.a X99 $30.00 Coyght 1999 by Academc Pess All ghts of eoducton n any fom eseved.

2 252 CABRELLI AND MOLTER and have been studed by Stchatz usng Foue and wavelet analyss Ž26, 28.. Amng to ecove self-smlaty aametes of hyscal sgnals, Hwang and Mallat study the self-smlaty of the wavelet tansfom Ž25.. One way to extend the noton of self-smlaty to functons, s to eque that the gah of the functon should be a self-smla set. If the functon s defned on a self-smla set, then we could eque that the functon shae n the self-smlaty of the doman;.e., f X wž X., then 1 f w Ž x. fž x., 1,...,n. Ž 1.1. Fo ths defnton we eque the w to be dsjont Ž.e., wž X. wž X. j, j.. Fom IFS-theoy t can be shown that f f s a contnuous functon satsfyng the self-smlaty condton Ž 1.1., f has to be constant. In ode to consde moe geneal solutons, we elax the condton of self-smlaty Ž 1.1., ntoducng a set of functons,..., and equng that f satsfy Ž. 1 n f w 1 Ž x. fž x., x w Ž X., 1,...,n. Ž 1.2. Fnally, to allow ovelang mas n the IFS, we ntoduce a functon O 1 that combnes the values of f w Ž x. fo 1,...,n fo the same x. In ths ae we study the exstence of self-smla functons n dffeent contexts and we elax even moe the self-smlaty condton Ž 1.2. allowng sace-deendent s and O. The oblem of fndng a functon u that satsfes a self-smlaty equaton of the tye, už x. O x, Ž u g.ž x.,..., Ž u g.ž x., Ž has been studed by Bajaktaevc n 1957 Ž2.. In the same yea, a smla equaton was consdeed by de Rahm Ž13., and condtons fo contnuous solutons wee found. In 24 Hutchnson, extendng the concet of selfsmlaty to aametc cuves, consdeed a atcula case of ths equaton. Related functonal equatons wee studed n factal nteolaton, n ode to show the exstence and constucton of contnuous factal functons Ž3, 6, 1517, 21, 22.. Cabell et al. n 9 constucted an oeato of the tye Ž 1.3. ntoducng a novelty to t: they added a set of gay-level functons, such that the esultng fxed ont of the oeato would no longe be stctly self-smla, but -self-smla. They woked n a atcula settng, n whch the functons had to satsfy vey estctve condtons to guaantee conve- gence.

3 GENERALIZED SELF-SIMILARITY 253 In ths ae we boaden the class of functons and we look at dffeent functonal saces and we ae able to emove most of the evous estctve condtons makng ths model much moe vesatle and theefoe moe sutable fo alcatons. We study the moe geneal equaton, Ž. už x. O x, x, Ž u g.ž x.,..., x, Ž u g.ž x., Ž that encloses most of the cases mentoned befoe and genealzes the concet of a self-smla functon Ž We fnd condtons on the comonents n ode to assue the exstence of solutons. We constuct an oeato on a sutable functon sace and the soluton of ou equaton s a fxed ont of ths oeato. Ths not only yelds a soluton of the equaton, but also shows that ths soluton can be comutatonally effcently calculated: we obtan t by teatng the oeato. Ths functonal equaton and the easy comutaton of ts soluton makes t sutable fo many alcatons. Fo examle, t models two stuatons whch ae of geneal nteest: usng factal comesson n mage o sgnal analyss and the constucton of wavelets and multwavelets. In the fst case, n sgnal ocessng, n atcula n mage eesentaton, a well-known oblem s the desgn of an adatve code fo a gven taget. Ths has been studed n atcula usng factals and self-smla models Žsee 1, 5, 7, 9, 12, 14, 20.. Some of the advantages of ths aoach ae the comesson ates acheved, and the comlexty of the mages that can be eesented. Geneally the stategy conssts n fndng an oeato T, whose fxed ont s the gven taget. In 12, t was shown that the evously ntoduced model Ž9. had the oety of beng dense, meanng that fo any functon and fo any one can constuct an oeato whose fxed ont s close than to the functon. Howeve, due to the estctons on the gay-level mas, ths esult was not enough fo actcal mlementatons. The functonal equaton consdeed n ths ae, eesents a genealzaton of the concet of self-smla functon extendng the alcablty of the model to a wde class of mages and allowng moe flexblty n the choce of the aametes. Ths should n tun lead to a bette comesson ate. Fo wok n ths decton we efe the eade to 8. In the second case, n the alcaton to the constucton of wavelets and multwavelets, one wants to fnd solutons to a efnement equaton of the tye, Ý Ž x. c Ž Ax k., Ž 1.5. kz d n ode to then constuct a wavelet decomoston of L 2 Ž R d.. Sutably defnng, g, and O n Ž 1.4. wll yeld Ž In the atcula case that all k

4 254 CABRELLI AND MOLTER the ck ae equal, one yelds the equaton studed by Gocheng and Madych n 19 and Stchatz n 27. Cuently thee s a gowng nteest n multwavelets, whch can be constucted usng efnement equatons n whch the coeffcents ae matces and the solutons ae vecto-valued functons Ž18, 23.. These matx efnement equatons ae atcula cases of ou functonal equaton, and solutons to these equatons, usng genealzed self-smla functons ae studed n Cabell, Hel, and Molte Ž10, 11.. In atcula the exstence of solutons to ths equaton n a sutable settng, lead to the constucton of one of the fst known examles of nonseaable othogonal multwavelets n R 2 Ž11.. We analyze two dffeent stuatons: n Secton 2 we study the case of bounded solutons wth the unfom metc. In Secton 3 we study L solutons fo 1. In both cases we gve suffcent condtons fo the exstence of solutons. 2. B X, E -CASE Let Ž X, d. be a comact metc sace and let Ž E, l. be a metc sace whee E s a closed subset of R m Žn atcula ŽE could be R m. and l could be a dstance n E nduced by some nom of R m. Let us also consde a ont t0 E that emans fxed thoughout the whole secton. We consde the functonal sace, BŽ X, E. u: X E, u bounded 4, wth DŽ u, v. su l už x., vž x., u, v BŽ X, E.. Ž 2.1. xx It s well known that Ž BŽ X, E., D. s a comlete metc sace. Let us now defne the functons O, w,, 1,..., n ode to constuct an oeato T on B X, E. Let O: X E E be nonexansve fo each x X,.e., ž / l O x, k, O x, k su l k, k, k, k E. Ž Let w : X X, 1,..., be njectve mas, whch ae not necessaly contactve, and let : X E E, 1,..., be functons that fo each x X satsfy the Lschtz condton, l Ž Ž x, k 1., Ž x, k2.. c l Ž k 1, k 2., k 1, k2 E, 1,...,, Ž 2.3. whee c 0 does not deend on x.

5 GENERALIZED SELF-SIMILARITY 255 In ode to be able to defne an oeato on BŽ X, E., we need some stablty condtons. We defne a functon f to be stable, f fž A. s bounded, wheneve A s a bounded set. Hence we assume that O and, 1,..., ae stable. Now we defne an oeato T on BŽ X, E. n the followng way, whee Ž. Ž Tu.Ž x. O x, x, u Ž x.,..., x, u Ž x. ; Ž už w 1 Ž x.., f x ImgŽ w., u Ž x. 1. Ž 2.5 ½. t 0, othewse, We use OŽ x, x, u x. fo the ght-hand sde of Ž We can ove the followng. THEOREM 2.1. Wth the eous notaton, f c max 1 c 0 s the Lschtz constant fo the s, then and T: BŽ X, E. BŽ X, E., DŽ Tu, Tv. cdž u, v.. In atcula, f c 1, T s contacte and theefoe thee exsts a unque u n B X, E such that Tu u. Poof. If u BŽ X, E. then t s easy to vefy that Tu BŽ X, E.. Now f u, v BŽ X, E. then ž Ž. Ž./ l Ž Ž Tu.Ž x., Ž Tv.Ž x.. l O x, Ž x, u Ž x.., O x, Ž x, v Ž x.. Theefoe, Ž. su l x, u Ž x., x, v Ž x. 1 su c l u Ž x., v Ž x. 1 c su l yx cdž u, v.. Ž už y., vž y.. DŽ Tu, Tv. cdž u, v.. We then have the followng.

6 256 CABRELLI AND MOLTER COROLLARY 2.2. If c 1, the functonal equaton, Ž. u O x, x, u Ž x.,..., x, u Ž x., Ž whee the u ae as n Ž 2.5., has a unque soluton n BŽ X, E.. The fxed ont of the oeato T s the soluton of the equa- Poof. ton. Note that Ž 2.6. s a genealzaton of the ognal functonal equaton gven n Ž In what follows, we study the oeato 2.4 n the L saces. 3. L -CASE Let now X R n comact, wth the n-dmensonal Lebesgue measue m and let E R wth some nom.. ŽNote: E could be chosen to be any Banach sace.. We consde the functons u: X E such that the eal-valued functon u Ž.. s Lebesgue-measuable, and, as usual, functons that ae equal almost eveywhee ae dentfed. If 1, let L Ž X, E. ½u: X E: H už x. dž x. 5, X wth u ŽH už x. dž x.. 1 ; and X 4 L X, E u: X E: u. essentally bounded, wth u Ž. ess.su. u.. It s well known, that L Ž X, E.,1 s a Banach sace. Let as befoe O: X E E be nonexansve,.e., ž / Ý O x, k O x, k k k. Ž 3.1. Fo measuable u: X E we defne as befoe the oeato 2.4, 1 1 Ž. Ž Tu.Ž x. O x, x, u Ž x.,..., x, u Ž x., 1 1 whee the ws and s ae as n the evous secton, wth the followng addtonal condtons: 1. The mas w 4 satsfy a Lschtz condton;.e., thee exst s 0, such that dw Ž Ž x,. wž y.. sdž x, y. whee d s the Eucldean dstance n R n. 2. The functons, 1,..., and O ae Boel measuable.

7 GENERALIZED SELF-SIMILARITY 257 These addtonal condtons ae equed n ode to guaantee the measuablty of Tu. We have the followng. PROPOSITION 3.1. Let T be defned as n the eous text, then Tu: X E s measuable fo each measuable functon u: X E and also f u, v ae measuable and u v a.e. then Tu Tv a.e. Poof. The measuablty of T u fo measuable u s a consequence of the stablty and the Boel-measuablty of O and the s and the fact that the w s ae Lschtz. Now f Z x: už x. vž x.4, then x: TuŽ x. 4 Tv x wž Z. 1. The Lschtz condton of the ws mles that ŽwŽ Z.. 0fŽ Z. 0 and theefoe the esult follows. Now we consde fst the sace L defned befoe. The case L 1 s teated late. THEOREM 3.2. Let T be the oeato of Pooston 3.1. Then, T: L L and Tu Tv c u v, u, v L. Poof. If u L then let Z X, Z 0 and u bounded n X Z. If we defne v: X E by v u X X Z, whee XXZ s the chaactestc functon of X Z, then v u a.e. and v s bounded. Then Tv s bounded and usng the ecedng ooston, Tu Tv a.e. and theefoe Tu L. Fom the oof of Theoem 2.1 we see that fo u and v L we have whch mles that Ž Tu.Ž x. Ž Tv.Ž x. cu v, a.e. on X, Tu Tv cu v. We wll now analyze the case L 1. We have the followng. THEOREM 3.3. Let T be the oeato of Pooston 3.1. Then, f u, v L Ž X, E., then Ž Tu Tv. L Ž X, E. and 1 n ž Ý / 1 Tu Tv sc u v, whee s and c ae the Lschtz constants of w and, esectely, and n s the dmenson of X. Futhemoe the fnteness of Ž X. yelds T: L Ž X, E. L Ž X, E..

8 258 Poof. and CABRELLI AND MOLTER If u, v L, then by Pooston 3.1, Tu Tv s measuable Tu Tv H Ž Tu.Ž x. Ž Tv.Ž x. dž x. X H Ž Ž.. Ž Ž.. X H Ý Ž. Ž. X 1 Ý H 1 X 1 1 Ý H Ž. Ž. 1 wž X. n Ý H 1 X n Ý 1 O x, x, u x O x, x, v x d x x, u x x, v x d x, by 3.1 c u x v x d x, by 2.3 c u w x v w x d x sc už t. vž t. dž t., Ž by the Lschtz condton of w. sc u v. Fom ths nequalty we see that f u, v L, then ž / 1 n Ý 1 Tv Tv Tu Tu sc u v Tu ; what says that f thee exsts a functon u L such that Tu L then T sends L nto L,1. Now, because X then L L,1 and because, by Theoem 3.2 T: L L, we get the desed esult. Ž n n COROLLARY 3.4. If, wth the ecedng notaton, Ý. 1sc 1 1 fo some, 1, then T s a contacton ma on L and the functonal equaton gen by 2.6, Ž. u O x, x, u Ž x.,..., x, u Ž x., 1 1 has a unque soluton n L. If the w : X X ae dffeentable and Dw Ž x. denotes the dffeental matx of w at the ont x, the oof of the last theoem shows that we

9 GENERALIZED SELF-SIMILARITY 259 can move the Lschtz oety of the oeato T, elacng s n by l su det Dw Ž x. s n. We then have the followng theoem. x X THEOREM 3.5. Let T be as defned by 3.1. Then, f u, v L Ž X, E., then and T: L Ž X, E. L Ž X, E., 1 ž Ý / 1 Tu Tv l c u v, whee l su det Dw Ž x. and c ae the Lschtz constants of. x X Note that the soluton to the functonal Eq. 2.6 esented hee can be obtaned as the lmt of the teaton of the oeato T at any statng functon. Remak. In 11 we show that usng the same technques than n ths ae, Theoem 3.5 can n some cases be slghtly moved weakenng the condtons on the. ACKNOWLEDGMENTS We wsh to acknowledge suot fom the Faculty of Mathematcs and the Deatment of Pue Mathematcs of the Unvesty of Wateloo. We thank Chs Hel fo many dscussons on the subject. REFERENCES 1. S. Abenda and G. Tuchett, Invese oblem of factal sets on the eal lne va the moment method, Nuoo Cmento 104B Ž 1989., M. Bajaktaevc, Su une equaton fonctonnelle, Glasnk Mat.-Fz. I Ast. 12Ž.Ž , M. F. Bansley, Factal functons and nteolaton, Const. Aox. 2 Ž 1986., M. F. Bansley, Factals Eveywhee, Academc Pess, San Dego, CA, M. F. Bansley, V. Evn, D. Hadn, and J. Lancaste, Soluton of an nvese oblem fo factals and othe sets, n Poceedngs of the Natonal Academy of Scences, Vol. 83, 1985, M. F. Bansley and A. N. Hangton, The calculus of factal nteolaton functons, J. Aox. Theoy 57 Ž 1989., M. F. Bansley and A. D. Sloan, A bette way to comess mages, BYTE Mag. Jan. Ž 1988., C. A. Cabell, M. C. Falsett, and U. M. Molte, Block-codng. A functonal aoach, ent, 1998.

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