Fractional Wavelet Transform in Terms of Fractional Convolution

Transkript

1 Progr. Frc. Differ. Appl., No. 3, Progress in Frcionl Differeniion nd Applicions An Inernionl Journl hp://dx.doi.org/0.2785/pfd/00305 Frcionl Wvele Trnsform in Terms of Frcionl Convoluion Ahilesh Prsd nd Prveen Kumr Deprmen of Applied Mhemics, Indin School of Mines, Dhnbd , Indi Received: 9 Oc. 204, Revised: 28 Feb. 205, Acceped: Mr. 205 Published online: Jul. 205 Absrc: Coninuiy of frcionl wvele rnsform FrWT in erms of frcionl convoluion operor nd is djoin re obined. A relion beween he FrWT nd inverse frcionl Fourier rnsform is esblished. The FrWT of es funcion spce is invesiged. Keywords: Frcionl wvele rnsform, Frcionl Fourier rnsform, Adjoin operor, Schwrz spce. Inroducion The word frcion is nowdys very populr in differen field of nowledge. We only menion he frcionl derivives in mhemics, he frcionl dimension in geomery nd frcionl rnsformions. In generl, i mens h some prmeer hs non-ineger vlue. The frcionl Fourier rnsform FrFT is generlizion of he convenionl Fourier rnsform wih n ngle θ. Mny yers go, i ws proposed in mhemics lierure bu ody mny new pplicions in severl res including Physics, Sochsic Process nd Mhemicl Anlysis re found [ 4. The one-dimensionl FrFT [5 9 wih n ngle θ of f L 2 R denoed byf θ fω ˆf θ ω is given by F θ fω ˆf θ ω K θ,ω fd R where C θ e i2 +ω 2 coθ 2 iω cscθ, θ nπ, K θ,ω 2π e iω, θ 2 π, δ ω, θ 2nπ, δ+ ω, θ 2n+π, n Z, δ denoes s he Dirc-del funcion nd C θ icoθ 2π. The inversion formul of FrFT is given by f K θ,ω ˆf θ ωdω, R where K θ,ω K θ,ω. Lemm Prsevl s Relion. If ˆf θ nd ˆψ θ re he FrFT of f nd ψ respecively, hen fψd Proof. See Ph e l. [9. ˆf θ ω ˆψ θ ωdω,. Corresponding uhor e-mil: pr bhu@yhoo.com 205 NSP Nurl Sciences Publishing Cor.

2 202 A. Prsd, P. Kumr: Frcionl Wvele Trnsform in Terms of... Definiion Ph [0 The Schwrz spce S is he se of rpidly decresing funcions φ C R such h γ α,β φsup α D β φ <, α,β N 0..2 R If f be loclly inegrble funcion onr. Then f generes disribuion in S s follows: f,φ fφd, φ S. R The spce SR is equipped wih he opology genered by he collecion of semi-norms {γ α,β }, i is Fréche spce. The dul of S is denoed by S nd is elemens re clled empered disribuions. Definiion 2 Ph e l. [9 The es funcion spce S θ is defined s: φ is member of S θ iff i is complex vlued infiniely differenible funcion onr, such h Γα,β θ φsup α β φ <, α,β N 0,.3 R where d d + i coθ. The FrWT ws inroduced firs by Mendlovic e l. [ s wy o del wih opicl signls. Shi e l. [2 inroduced he coninuous ffine rnsformion nd chirp modulion of moher wvele ψ L 2 R s ψ b,,θ b ψ e i 2 2 b 2 coθ, for ll >0, b R nd θ s bove nd defined novel FrWT of squre inegrble funcion. If θ π 2, hen ψ b,,θ reduces o convenionl moher wvele defined in [0, 3 5. Prsd e l. [6 defined he FrFT of ψ b,,θ is given by ˆψ b,,θ θ ω [ e 2 ib2 +ω 2 coθ ibω cscθ 2 i 2 ω 2 coθ F θ e i 2.2 coθ ψ ω,.4 nd esblished generlized coninuous frcionl wvele rnsform of funcion f L 2 R. The coninuous frcionl convoluion of wo coninuous funcions f,ψ L 2 R is defined s f θ ψ fξψ ξ e i 2 2 ξ 2 coθ dξ,.5 where θ is nown s he coninuous frcionl convoluion operor. The wvele rnsform ssocied wih convenionl Fourier rnsform ws sudied in [0, 4, 5 nd corresponding FrWT involving FrFT ws invesiged in differen wy [, 2, 6, 7. The squre inegrble boundedness resuls for he wvele rnsform nd is djoin were proved nd esblished relion beween wvele nd Fourier rnsform [8. In his pper coninuiy of FrWT in erms of frcionl convoluion operor nd is djoin re obined. A relion beween he FrWT nd inverse FrFT is esblished. The frcionl wvele of es funcion spce is invesiged. 2 Coninuiy of frcionl wvele rnsform The coninuous FrWT of f L 2 R w.r.. he wvele ψ L 2 R is defined [, 2, 6 s: W ψ θ f b, f θ b, f ψ b,,θ d. 2. Using. of Prsevl s ideniy nd.4, i follows from 2. s: W ψ θ f b, f θ b, [ e i 2 b2 +ω 2 coθ+ibω cscθ+ 2 i 2 ω 2 coθ F θ e i 2.2 coθ ψ ω ˆf θ ω dω. 2.2 Definiion 2 Prsd e l. [6 A wvele ψ L 2 R nd sisfies he following dmissibiliy condiion F θ e i 2.2 coθ ψ v 2 C ψ,θ dv<, 2.3 v where F θ denoes he FrFT operor. 205 NSP Nurl Sciences Publishing Cor.

3 Progr. Frc. Differ. Appl., No. 3, / Lemm 2 The coninuous frcionl convoluion rnsform of funcion f L 2 R w.r.. he funcion ψ L 2 R b depending on, is denoed bywb θ f nd defined s W θ b f f θ ψ b. Then W θ ψ fb, W θ b f, 2.4 where ψ ψ. Proof. Using.5, we hve W θ b f f ψ b e i 2 b2 2 coθ d f ψ b f ψ b,,θ d W θ ψ f b,. e i 2 2 b 2 coθ d Remr 2 Using 2.2, hen from Lemm 2 we hve [ Wb θ f e i 2 b2 +ω 2 coθ+ibω csc θ+ 2 i 2 ω 2 coθ F θ e i 2.2 coθ ψ ω ˆf θ ω dω. 2.5 As per [8, we hve define he operorw θ b s: Definiion 22 The operorwb θ ssocied wih Wb θ is defined by [W θ b f f ψ b,,θ d b f ψ e i 2 2 b 2 coθ d. 2.6 Lemm 22 If f,g L 2 R, hen he operorw θ b is djoin operor of W θ b. Proof. Le f,g L 2 R. Define W θ b g, f nd g,w θ b f f Wb θ gd b f g ψ Wb θ g, f. e i 2 2 b 2 coθ d d, g [Wb θ fd b g f ψ e 2 i2 b 2 coθ d d b f g ψ e 2 i 2 b 2 coθ d d This complees he proof of Lemm. Lemm 23 If frcionl convoluion operor W θ b is self-djoin, hen he generlized frcionl wveles ψ b,,θ re given by ψ b,,θ / ψ b,,θ, R\{0} NSP Nurl Sciences Publishing Cor.

4 204 A. Prsd, P. Kumr: Frcionl Wvele Trnsform in Terms of... Proof. From Lemm 2 nd Definiion 22, i follows h operor W θ b W θ b f[w θ b f. Now is self-djoin if Wb θ f W θ ψ b, f b f ψ e 2 i 2 b 2 coθ d. 2.8 Similrly [W θ b f f ψ b,,θ d b f ψ From 2.8 nd 2.9, we hve he desired resul. e i 2 2 b 2 coθ d. 2.9 Remr 22 If seing x b/ nd β b/, we find h operor W θ b will be self-djoin for ψ defined by ψxβ/x ψβ/x b/e i 2 2 [β/x 2 coθ, x R\{0}. 2.0 This requires he rio b/ o be consn. Furher, 2.7 holds if. The following heorem yields he L 2 -coninuiy of he operor W θ b ndw θ b. Theorem 2 If ψ sisfies he dmissibiliy condiion 2.3. Then djoin W θ b L 2 R nd W θ b f 2 C ψ,θ f 2. Similrly, if W θ b is self-djoin operor, hen W θ b f 2 C ψ,θ f 2. Proof. Firs, we shll show h Wb θ f L 2 R nd Wb θ f 2 C ψ,θ f 2. For ny h L 2 R using Schwrz inequliy, we hve [Wb θ fh d h [Wb θ f d h f ψ b,,θ d d f h ψ b,,θ d d Now, using Prsevl s relion, we hve [W θ b f h C ψ,θ h 2 f 2. f W θ b h d /2 f 2 d Wb θ h 2 d /2 f 2 d [ F θ e i 2.2 coθ ψ ωĥ θ 2 ω dω d [ [ f 2 F θ e i 2.2 coθ ψ /2 C ψ,θ f 2 ĥ θ ω dω 2. /2 e i 2 b2 +ω 2 coθ+ibω cscθ+ i 2 2 ω 2 coθ /2 /2 ω 2 d ĥ θ ω 2 dω By converse of Schwrz inequliy [9, p. 385, if h L 2 R nd [W θ b f h C ψ,θ h 2 f 2, hen we ge desired resul. Anologously, we cn prove he second pr. 205 NSP Nurl Sciences Publishing Cor.

5 Progr. Frc. Differ. Appl., No. 3, / Relion beween he frcionl wvele rnsform nd inverse frcionl Fourier rnsform Le mesurble funcion ψ defined on R which sisfies ψv v /2 dva<, insed of he dmissibiliy condiion Theorem 3 If Eb, θ e i 2 2 coθ+ibcscθ, ψ L R nd ψv v /2 dv<. Then E θ b, W θ b F θ signf θ [W θ b E θ b,, where sign {, >0, <0. Proof. Le f L 2 R nd gf θ ˆf θ. Then y b Wb θ g ψ e 2 i y2 b 2 coθ gydy y b ψ e 2 i y2 b 2 coθ K θ y, ˆf θ d dy ψv e 2 i [v+b2 b 2 coθ K θ v+b, ˆf θ d dv sign K θ b, ˆf θ e ivcscθ ψvdv d sign K θ b, ˆf θ e iucscθ ψu/du d sign e 2 i 2 coθ ibcscθ K θ x b x, ψ e 2 i x2 b 2 coθ ˆf θ e i 2 2 coθ+ibcsc θ d dx Therefore sign e 2 i 2 coθ ibcscθ W θ b F θ ˆf θ sign e i 2 2 coθ ibcscθ or K θ x, ψ b,,θ x ˆf θ e i 2 2 coθ+ibcscθ d dx. 3.2 K θ x,[w θ b ˆf θ e i 2 2 coθ+ibcscθ xdx, e i 2 2 coθ+ibcscθ W θ b F θ ˆf θ signf θ [W θ b ˆf θ e i 2 2 coθ+ibcsc θ. Hence E θ b, W θ b F θ signf θ [W θ b E θ b,. Theorem 32 Le ψ be periodic funcion wih period b/ defined on[ σ, σ, where σ mx, β, nd sisfies ψx x /2 dx<. x Define ψxβ/x ψβ/x b/e i 2 2 [β/x 2 coθ for x >σ. Then E θ b, W θ b F θ signf θ [W θ b Eθ b,. 205 NSP Nurl Sciences Publishing Cor.

6 206 A. Prsd, P. Kumr: Frcionl Wvele Trnsform in Terms of... Proof. Using 2.0 for β <, we hve ψv v /2 dv ψβ/v b/e 2 i 2 [β/v 2 coθ β/v v /2 dv v v β /2 x b ψ x /2 dx x β β /2 ψx x /2 dx<. x Now gin, using 2.0 for β, we hve ψ /2 d ψβ/ b/e 2 i 2 [β/ 2 coθ β/ /2 d β β β /2 x b ψ x /2 dx x β /2 ψx x /2 dx<. x Since Wb θ is self-djoin, so by using 3.2 nd 2.7, we ge Wb θ g sign e 2 i 2 coθ ibcscθ K θ x, sign e i 2 2 coθ ibcscθ K θ x,[w θ b x ψ b,x,θ ˆf θ e i 2 2 coθ+ibcscθ d ˆf θ e i 2 2 coθ+ibcscθ xdx. Therefore e i 2 2 coθ+ibcscθ W θ b F θ ˆf θ signf θ [W θ b ˆf θ e i 2 2 coθ+ibcscθ. Hence E θ b, W θ b F θ signf θ [W θ b Eθ b,. dx 4 Frcionl wvele rnsform of generlized funcions As per [8, 20, we define he frcionl wvele rnsform of ψ,φ S θ R by Wψ θ b f b, f ψ e 2 i 2 b 2 coθ d. Definiion 4 Ph [8 A complex vlued smooh funcion ηb, belongs o he spce BR 2 if nd only if γ l,n b l m+n m, η sup n + b + ηb, <, 4. b, R 2 for ll l,n,m, N 0. Lemm 4 Le ψ b,,θ be wvele. Then m ψb,,θ d m [ ψ b, e 2 i2 b 2 coθ, m N 0, d where d d i coθ. Proof. The proof is very esy nd lef o he eger reder. Lemm 42 If f, φ S θ R. Then [ f φ where r0 r f A,r r r φ, N 0, is he sme defined in.3 nd A,r re consns. 205 NSP Nurl Sciences Publishing Cor.

7 Progr. Frc. Differ. Appl., No. 3, / Proof. Since f,φ S θ R. Then [ f φ + i coθ fφ Similrly, f φ f φ i coθ fφ f φ f + i coθ φ f φ+ f φ r0 2 [ f φ r f A,r r [ r φ. f.φ [ f + i coθ 2 f 2 φ 2 f 2 r0 r f A 2,r r In generl, for N 0, we ge [ f φ r0 r f A,r r φ+ f [ φ f φ φ+ f 2 φ 2 r φ. r φ. Remr 4 If f,φ S θ R. Then [ fφ r0 r f B,r r + i coθ [ f. φ r φ, N 0, 4.2 where is he sme s defined in Lemm 4 nd B,r re consns. Lemm 43 If wvele ψ b,,θ is differenible. Then + ψb,,θ b ψb,,θ, N 0, where is he sme s defined in Lemm 4. Proof. We hve + ψb,,θ b + ψ [ b 2 ψ e 2 i2 b 2 coθ + b ψ e 2 i2 b 2 coθ b 2 ψ b e 2 i 2 b 2 coθ ψb,,θ b. e i 2 2 b 2 coθ b 3 ψ b e 2 i2 b 2 coθ 205 NSP Nurl Sciences Publishing Cor.

8 208 A. Prsd, P. Kumr: Frcionl Wvele Trnsform in Terms of... Similrly, 2 + ψb,,θ + + ψb,,θ [ + ψb,,θ b b [ b 2 ψ 3 ψ b ψb,,θ. b b 2 2 In generl, by inducion on N 0, we ge + ψb,,θ b ψb,,θ. e i 2 2 b 2 coθ e 2 i2 b 2 coθ + ψ b,,θ Remr 42 If ψ,φ S θ R nd N 0. Then b φ ψb,,θ d [φb ψ b,,θ d. 4.3 Theorem 4 If moher wvele ψ S θ R. Then he FrWT W θ ψ is coninuous liner mpping of S θr ino BR 2. Proof. Using Lemm 4, 42 nd 43 nd inegring by prs, we hve + Wψ θ φb, + φ b ψ e 2 i2 b 2 coθ d φ b ψb,,θ d Now m b + Wψ θ φb, Similrly m+n b + Wψ θ φb, [φ b m+n+ m+ [φ b Now, using he inequliy b l b + l l 2 l b ψ b,,θ d. m+ [φ b m [ ψ b, e 2 i2 b 2 coθ d m+ [φ b m m+ m+ [φ b n [ ψ b, e 2 i 2 b 2 coθ d. l + /, l l > 0, ψb,,θ d ψ b,,θ d. 205 NSP Nurl Sciences Publishing Cor.

9 Progr. Frc. Differ. Appl., No. 3, / we cn wrie m+n bl n b + Wψ θ φb, [ 2 l b l 2 l b l + / l m+ [φ b ψn A,m,r b r r0 +2 l / l A,m,r b r r0 + r0 r0 r0 m+ r φ m+ r φ A,m,r 2 l sup b r m+ r φ R A,m,r 2 l sup l b r m+ r φ R A,m,r 2 l[ + l sup b r R +sup l b r m+ r φ R b, R 2 b b ψn d b ψn d b l b ψn d b ψn d m+ r φ ψ n u du. Therefore b l m+n sup n + b + Wψ θ φb, r0 r0 u l ψ n u du e i 2 2 b 2 coθ A,m,r 2 l[ Γ r,m+ r φ u l ψ n u du+ Γ l, r,m+ r φ ψ n u du A,m,r 2 l[ Γ r,m+ r φsup +u 2 u l D n ψu u R +Γ l, r,m+ r φsup +u 2 D n ψu u R +u 2 du, +u 2 du where A,m,r > 0. Since φ,ψ S θ R ech erm of he righ-hnd side is convergen. Hence, Wψ θ is coninuous liner mpping of S θ R ino BR 2. d 5 Exmples In his secion we shll illusre, by mens of some exmples, he dvnges of he frcionl wvele rnsform in erms of frcionl convoluion s: If ψe 2 i 2 coθ nd φ 2π 2 e icoθ. Then φ θ ψ 0 φξψ ξe i 2 2 ξ 2 coθ dξ n 3 2 θ 2 coθe 2 2 coθ, which is frcionl Mexicn h wvele. If θ π/4, hen i reduce o he convenionl Mexicn h wvele. [ Similrly if ψe 2 i 2 coθ nd φ 2π e 2 +ξ 0 2 +i 2 coθ. Then φ θ ψ n 2 θ e [iξ coθ, which is frcionl Morle wvele. If θ π/4, hen i reduce o he convenionl Morle wvele. 205 NSP Nurl Sciences Publishing Cor.

10 20 A. Prsd, P. Kumr: Frcionl Wvele Trnsform in Terms of... Acnowledgemen This wor is sponsored by NBHM DAE projec, Gov. of Indi, under grn No. 2/484/20/-R&D II/350. The uhors re very hnful o he referees for heir vluble commens nd suggesions o improve he pper. References [ H. M. Ozs, Z. Zlevsy nd M. A. Kuy, The Frcionl Fourier Trnsform wih Applicions in Opics nd Signl Processing, New Yor: Wiley, [2 X. G. Xi, On bndlimied signls wih frcionl Fourier rnsform, IEEE Signl Process. Le. 3 3, [3 E. Scjdic, I. Djurovie nd L. Snovic, Frcionl Fourier rnsform s signl processing ool: An overview of recen developemens, Signl Process. 9, [4 T. Aliev, M. J. Bosins nd M. L. Clvo, Frcionl cyclic rnsforms in opics: heory nd pplicions in Recen Reserch nd Developmens in opics, S. G. Pndli, Ed., Reserch Signpos, Trivndrum, Indi,, [5 L. B. Almeid, The frcionl Fourier rnsform nd ime-frequency represenions, IEEE Trns. Signl Process. 42, [6 F. H. Kerr, Nmis frcionl Fourier rnsform on L 2 nd pplicions o differenil equions, J. Mh. Anl. Appl. 36, [7 A. I. Zyed, Frcionl Fourier rnsform of generlized funcions, Inegrl Trnsforms Spec. Func. 7 3, [8 A. Prsd nd M. Kumr, Produc of wo generlized pseudo-differenil operors involving frcionl Fourier rnsform, J. Pseudo- Differ. Oper. Appl. 2 3, [9 R. S. Ph, A. Prsd nd M. Kumr, Frcionl Fourier rnsform of empered disribuions nd generlized pseudo-differenil operor, J. Pseudo-Differ. Oper. Appl. 3 2, [0 R. S. Ph, The Wvele Trnsform, Vol. 6, Alnis Press/World Scienific, Pris, Frnce, [ D. Mendlovic, Z. Zlevsy, D. Ms, J. Grci nd C. Ferrei, Frcionl wvele rnsform, Appl. Op , [2 J. Shi, N. Zhng nd X. Liu, A novel frcionl wvele rnsform nd is pplicions, Sci. Chin Inf. Sci. 55 6, [3 I. Ruiz nd B. G. Zpirin, Improvemen of Shimmer prmeer of Oesophgel voices using wvele rnsform: Wvele rnsforms nd heir recen pplicions in biology nd geoscience, Edied by D. Blenu, Publisher: InTech 202, DOI: /2286. [4 I. Dubechies, Ten Lecures on Wveles, CBMS-NSF Regionl Conference Series in Applied Mhemics, SIAM Publ., Phildelphi, PA, [5 C. K. Chui, An Inroducion o Wveles, Acdemic Press, USA, 992. [6 A. Prsd, S. Mnn, A. Mho nd V. K. Singh, The generlized coninuous wvele rnsform ssocied wih he frcionl Fourier rnsform, J. Comp. Appl. Mh. 259, [7 A. Prsd nd A. Mho, The frcionl wvele rnsform on spces of ype S, Inegrl Trnsforms Spec. Func. 23 4, [8 R. S. Ph, Coninuiy nd inversion of he wvele rnsform, Inegrl Trnsforms Spec. Func. 6-4, [9 R. R. Goldberg, Cerin operors nd Fourier rnsform on L 2, Proc. Amer. Mh. Soc. 0, [20 M. Holschneider, Wvele-An Anlysis Tool, Clrendon Press, Oxford, NSP Nurl Sciences Publishing Cor.