Wavelet Transform of Fractional Integrals for Integrable Boehmians

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1 Available a hp://pvamu.edu/aam Appl. Appl. Mah. ISSN: Vol. 5, Issue (Jue 200) pp. 0 (Previously, Vol. 5, No. ) Applicaios ad Applied Mahemaics: A Ieraioal Joural (AAM) Wavele Trasorm o Fracioal Iegrals or Iegrable Boehmias Desha Looer ad P. K. Baerji Deparme o Mahemaics Faculy o Sciece J. N. V. Uiversiy Jodhpur , Idia deshap@yahoo.com baerjip@yahoo.com S. L. Kalla Deparme o Mahemaics & Compuer Sciece Kuwai Uiversiy Saa 3060, Kuwai shyamalla@yahoo.com Received: April 25, 2009; Acceped: March 23, 200 Absrac The prese paper deals wih he wavele rasorm o racioal iegral operaor (he Riema- Liouville operaors) o Boehmia spaces. By virue o he exisig relaio bewee he wavele rasorm ad he Fourier rasorm, we obaied iegrable Boehmias deied o he Boehmia space or he wavele rasorm o racioal iegrals. Keywords: Wavele rasorm, Fourier rasorm, Riema-Liouville racioal iegral operaors, disribuio spaces, Boehmia, Iegrable Boehmias MSC 200 No: 65T60, 44A38, 26A33, 46F2, 46F99

2 2 Looer e al.. Iroducio Wavele is a ew area ha sads a he iersecio o roiers o mahemaics, scieiic compuig ad sigals ad image processig. I has bee oe o he major research direcios i sciece i he las decade ad is sill udergoig rapid growh. Some group o mahemaicia view i as a ew basis or represeig ucio; some cosider i as a echique or ime requecy aalysis. Wavele aalysis is a absrac brach o mahemaics ha is origiaed as a lac i Fourier aalysis. I order o elimiae he weaess o idig he requecy specrum o a sigal locally i ime, Gabor (946) irs iroduced he Widowed Fourier rasorm (or shor ime Fourier rasorm) or Gabor rasorm by usig a Gaussia disribuio ucio as he widow ucio Gabor (946). The cocep o waveles or odelees sared o appear i he lieraure oly i early i 980s. Morle e al. (982) iroduced he idea o wavele rasorm as a ew ool or seismic sigal aalysis. Grossma broadly deied waveles i he coex o quaum physics. The by he joi veure o mahemaical group i Marseilles, led by Grossma, i collaboraio wih Daubechies, Paul ad ohers, exeded Morle discree versio o wavele rasorm o he coiuous versio by relaig i o he heory o cohere saes i quaum physics. See Grossma ad Morle (984), ad Daubechies (992, 998a, 998b). Meyer lear abou he wor o Morle ad Marseilles group ad applied he Lile-wood Paley heory o he sudy o wavele decomposiio Meyer (986), where he also explais he cosrucio o waveles ad he applicaio o wavele series represeaio o he aalysis or ucio spaces such as Hölder, Hardy, Besov ad sudied he oio o holomorphic waveles. Crediable coribuios o he wavele heory is made by may auhors Chui (992), Daubechies (992, 998a), Grossma ad Morle (984), Jasee (98a, 98b), Miusińsi (983, 988). The Gabor rasorm (i.e., he widowed Fourier rasorm) o wih respec o g [c. Debah (998, p.688)] is G [ ](, ) i g e ( ) ( ) d 2 () (, g, ), 2 (2) where, g L 2 ( R) wih he ier produc (, g). For a ixed, ~ G (, ) (, ) F{ ( )} ˆ ( ), (3) g

3 AAM: Ier. J., Vol. 5, Issue (Jue 200) [Previously, Vol. 5, No. ] 3 where F is he Fourier rasorm ad G is he Gabor rasorm. Parseval ormula or Gabor rasorm is give by ~, g ~ g 2, h, (4) whereas he iversio ormula is i ( ) (, )) ( ). 2 g g e dd 2 g (5) Basic properies o he Gabor rasorm are lieariy, raslaio, modulaio, ad cojugaio. The heory o Gabor rasorm has bee geeralized by Jasse (98a, 98b) or empered disribuios S. The reame o wavele rasorm wih he Schwarz disribuio was explaied by auhors Padey (999), Paha (998), Waler (992), Waler (993), Waler (994), amog ohers. I Baerji e al. (2004), auhors ivesigae he Gabor rasorm or iegrable Boehmias. Deiiio ad ermiologies, releva o prese wor ad he covergece or he Boehmia space are explaied laer. The racioal calculus (racioal iegrals ad derivaives, also called racioal dieriegrals) have several applicaios i iegral rasorms ad disribuio spaces, which are called racioal rasorms, see or isace, Looer ad Baerji (2007). I he prese wor, usig relaio bewee he Fourier ad he Wavele rasorm, we have obaied he Gabor rasorm or racioal iegral operaor (Riema-Liouville ype) which is urher proved or iegrable Boehmias. Deiiio. Samo e al. (993, p. 33): Le ( x) L ( a, b). The he iegrals x ( I a)( x) ( x ) ( ) d, x a, ( ) a b ( I b)( x) ( x) ( ) d, x b, ( ) x (6) (7) where 0, are Riema-Liouville racioal iegrals o order. They are also ow as lesided ad righ-sided racioal iegrals, respecively. Ideed, hese iegrals are exesios rom he case o a iie ierval [ a, b] o he case o a hal-axis, give by

4 4 Looer e al. ( ) (8) x ( I0)( x) ( x) ( ) d, 0 x, 0 while or he whole axis, i is give, respecively, by Samo e al. (993, p. 94) as ad ( ) (9) x ( I )( x) ( x) ( ) d, x, ( ) (0) x ( I )( x) ( x) ( ) d, x. The covoluio o ormulae (9) ad (0) is ( I )( x) ( x ) d, ( ) where ( ) 0 ( x ) d, () ad, 0, 0, 0, (2) 0, 0,, 0. (3) Fracioal iegral or ucio L (, ), 0 ad p /, is give by p ( I )( x) ( x ) d ( x ) d. ( ) 0 ( ) (4) There are wo ways o deie he racioal iegrals ad derivaives o geeralized ucios Samo e al. (993, p. 46). The irs is based o he deiiio o a racioal iegral operaor as a covoluio

5 AAM: Ier. J., Vol. 5, Issue (Jue 200) [Previously, Vol. 5, No. ] 5 ) ( x ( ), (5) o he ucio ( ) x, wih he geeralized ucio. The secod is by virue o he use o adjoi operaors. By employig racioal iegraio by pars, ormulae (6) ad (7) assume he orm ( I )( ) (, I )( ). (6) a b The ucio, i (6) may, ideed, be deied as he geeralized ucio i I b maps coiuously he space o es ucios X io isel. Whe ad I a ( ) are cosidered o be geeralized ucios o diere spaces o es ucio X ad Y such ha X (he dual o he es ucio space X) ad X coiuously. The racioal iegraio I a Y ( ) (he dual o he es ucio space Y), I o a geeralized ucio I b maps Y io (he dual o ) is give by I, ) (, I, ),. (7) ( Ideed, usig (7), he Fourier rasorm is give by ~ ~ (, I ) (, I ) (,( ix) ˆ( x)), (8) which is derived by virue o he oios o covoluio prescribed i (6). The Fourier rasorm o he racioal iegrals I are [Samo e al. (993, p. 47)] F( I ) ( ix) ˆ( x), L ( a, b). (9) Sudy o regular operaors o Miusińsi by Boehme (973) resuled io he heory o Boehmias, he geeralizaio o Schwarz disribuio heory. These regular operaors orm subalgebra o Miusińsi operaors such ha hey iclude oly such ucios whose suppor is bouded rom he le, ad a he same ime do o have ay resricio o he suppor. The geeral cosrucio o Boehmias gives rise o various ucio spaces, which are ow as Boehmia spaces [c. Miusińsi ad Miusińsi (98) ad Miusińsi (983, 988)]. I is observed ha hese spaces coai all Schwarz disribuios, Roumieu ulradisribuios ad empered disribuios. The ame Boehmia is used or all objecs by a algebraic cosrucio, which is similar o he cosrucio o he ield o quoies. Suppose G is a addiive commuaive semigroup, S be a

6 6 Looer e al. subse o group G such ha S G is a sub semigroup, or which we deie a mappig rom G S o G such ha ollowig codiios are saisied (hese codiio are or he mappig ): (i) i, S he ( ) S ad (ii) i G,, S he ( ) ( ) (iii) i, G, S he ( ) ( ) ( ). The dela sequece, deoed by, is deied as members o class dela which are he sequeces o subse S, ad saisies he codiios (i) i G,( ) ad, he i G., (ii) i ( ),( ) he ). ( The he quoie o sequeces is deied as he eleme o cerai class A o pair o sequeces deied by N A{( ),( ):( ) G,( ) }. This is deoed by / such ha, m, N. m m Furher, he quoies o sequeces g, N. / ad g are called equivale i The equivalece relaio deied o A ad he equivalece classes o quoie o sequece are called Boehmias. The space o all Boehmias, deoed by B, has he properies addiio, muliplicaio ad diereiaio. The Boehmia space BL will be called he space o locally iegrable Boehmias i he group G be he se o all locally iegrable ucio o R ad possibly wo such ucios are ideiied wih respec o Lebesgue measure ( hese ucios are equal almos everywhere) ad he opology o his space is ae o be he semi-orm opology geeraed by p ( ) d,, 2, 3,,

7 AAM: Ier. J., Vol. 5, Issue (Jue 200) [Previously, Vol. 5, No. ] 7 where is he usual Lebesgue measure o R ad D( R). I oher words, i L ad ( ) is he dela sequece, he ( ) 0, as. A pair o sequeces (, ) is called a quoie o sequeces, ad is deoed by /, i L (,2, ), where ( ) is a dela sequece ad m m, m, N. Two quoies o sequeces / ad g / are equivale i g, N. The equivalece class o quoie o sequeces will be called a iegrable Boehmai, he space o all iegrable Boehmia will be deoed by B L. Covergece o Boehmias is deied i Miusińsi (983). The ermiologies regardig Boehmias ad Boehmia spaces ca be reerred o i Miusińsi ad Miusińsi (98), Miusińsi (983, 988). We remar ha prese ivesigaios are idepede o he resuls give i Bargma (96, 967). 2. Wavele Trasorm o Fracioal Iegrals or Iegrable Boehmias Usig he relaio bewee he Gabor ad he Fourier rasorm, relaios (3) ad (9), respecively, he racioal iegrals or he Gabor rasorm, ca be wrie i he orm F ( I ( )) ( i) ˆ ( ), L ( a, b). (20) I oher words, (20) ca be wrie as i.e., G ( I ) ( i ) ˆ ( ), (2) G ( I ) ( i) ( ˆ ( )) ( i ) ( ˆ ) ( ). (22) Theorem : I[ / ] BL, he he sequece G ) ( i ) ( ˆ ) ( ) (23) ( I coverges uiormly o each compac se i R.

8 8 Looer e al. Proo: I ( ) is a dela sequece, he ( ˆ ) coverges uiormly o each compac se o he cosa ucio uiy. Thereore, ( ˆ ) 0 o K (he compac se) ad, hus, he le had side o (23) gives ˆ ˆ ( ˆ )( ˆ ( I )( ) ( I )^ I ) G ( I ) o K ( ˆ ) ( ˆ ) ( ˆ ) ( i) ( ˆ ) ( ˆ ), [c. Equaio (22)]. ( ˆ ) This shows ha he Gabor rasorm o racioal iegrals or a iegrable Boehmia F [ / ] ca be expressed as he limi o he sequece G ( I ), which, i ac, is he space o all coiuous ucios o R. This proves he heorem compleely. Propery : Le [ / ] BL. The, lim F F, G ( I F ) G( I F) uiormly o each compac se. Proo: We have lim F F G{ F } G{ F}, uiormly o each compac se. The sequece ca be expressed as F F L,, N, which has a orm, ( F F) 0, as, N, where K is well deied. Sice G{ } is a coiuous ucio, we have G{ } 0 o K or N. I is, hereore, eough o prove ha G F } G{ } G{ F} G{ }, { uiormly o K. We have, G F } G{ } G{ F} G{ } G{( F F) }, {

9 AAM: Ier. J., Vol. 5, Issue (Jue 200) [Previously, Vol. 5, No. ] 9 such ha ( F F) 0, as. This jusiies he exisece ad validiy o he propery. 3. Coclusios The prese paper ocuses o he applicaio o he Riema Liouville ype racioal iegral operaor o he Gabor rasorm ad he iegrable Boehmias. The racioal iegral ormula or he Gabor rasorm is give by usig he relaio bewee he Gabor ad he Fourier rasorms. The ormula ad he propery esablished i his paper are suiable or cerai Boehmia space or a iegrable Boehmia. The compac se ad he coiuiy o he ucio used, approves he exisece o he resuls give i his paper. Acowledgeme The prese wor was iiially suppored by he Pos Docoral Fellowship o NBHM (DAE), Idia, Sacio No. 40/7/2005-R&D II/37. The revisio o he paper i is prese modiied orm is suppored uder DST (SERC) Youg Scieis Fas Trac Scheme, Goverme o Idia, boh o which are sacioed o he irs auhor (DL). The secod auhor (PKB) acowledges he UGC (Idia) or awardig he Emerius Proessorship, uder which his prese orm o he paper is coceived. The auhors are haul o he reviewers/reerees or ruiul criicism ad commes or he improveme. REFERENCES Baerji, P. K., Looer, Desha ad Debah, Loeah (2004). Wavele rasorm or iegrable Boehmais, J. Mah. Aal. Appl., Vol. 296, No. 2, pp Bargma, V. (967). O a Hilber Space o Aalyic Fucios ad a Associaed Iegral Trasorm (II): A Family o Relaed Fucio Spaces Applicaio o Disribuio Theory, Comm. Pure Appl. Mah., Vol. 20, pp. -0. Bargma, V. (96). O a Hilber Space o Aalyic Fucios ad a Associaed Iegral Trasorm (I), Comm. Pure Appl. Mah., Vol. 4, pp Boehme, T. K. (973). The suppor o Miusińsi Operaors, Tras. Amer. Mah. Soc., Vol. 76, pp Chui, C. K. (992). A Iroducio o Waveles, Sa Diego, CA, Academic Press, Sa Diego, CA. Daubechies, I. (992). Te Lecures o Waveles, Sociey or Idusrial & Applied Mahemaics, Philadelphia.

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