DUNKL WAVELETS AND APPLICATIONS TO INVERSION OF THE DUNKL INTERTWINING OPERATOR AND ITS DUAL

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1 IJMMS 24:6, PII. S Hindwi Publishing Corp. UNKL WAVELETS AN APPLICATIONS TO INVESION OF THE UNKL INTETWINING OPEATO AN ITS UAL ABELLATIF JOUINI eceived 28 ecember 22 We define nd study unkl wvelets nd the corresponding unkl wvelets trnsforms, nd we prove for these trnsforms Plncherel nd reconstruction formuls. We give s ppliction the inversion of the unkl intertwining opertor nd its dul. 2 Mthemtics Subject Clssifiction: 42C15, 44A Introduction. We consider the differentil-difference opertor Λ α on introduced by unkl [1] nd clled in the literture unkl opertor on of index α+1/2 ssocited with the reflection group Z 2,givenby Λ α ux = dux dx + α+1/2 [ ] 1 ux u x, α> x These opertors re very importnt in mthemtics nd physics. They llow the development of generlized wvelets from generlized continuous clssicl wvelet nlysis. Moreover,we hve provedin [2] tht the generlized two-scle eqution ssocited with the unkl opertor hs solution nd then we cn define continuous multiresolution nlysis. unkl hs proved in [1] tht there exists unique isomorphism V α, clled the unkl intertwining opertor, from the spce of polynomils on of degree n onto itself, stisfying the trnsmuttion reltion d Λ α V α = V α dx, 1.2 V α 1 = ösler hs proved in [3] tht for ech x there exists probbility mesure µ x on with support in the intervl [ x,x], such tht for ll polynomils p on, we hve V α px = pydµ x y. 1.4 Next, Trimèche in [5] hs extended the opertor V α to n isomorphism from, the spce of C -functions on, onto itself stisfying the reltions 1.1 nd 1.2, nd hs shown tht for ech x, there exists unique distribution η x in, the spce of distributions of compct support on, with support in the intervl [ x,x] such

2 286 ABELLATIF JOUINI tht Vα 1f x = ηx,f, f. 1.5 He hs shown lso in [5] tht the trnsposed opertor t V α of the opertor V α hs the integrl representtion y, t V α f y = fxdν y x, f, 1.6 where ν y is positive mesure on with support in the set {x : x y } nd f in, the spce of C -functions on with compct support, nd t V α is n isomorphism from onto itself, stisfying the reltions y, t V α Λα f y = d t V α f y, 1.7 dy nd for ech y, there exists unique distribution Z y in, the spce of tempered distributions on, such tht t 1f V α y = Zy,f, f. 1.8 In this pper, we re interested in unkl wvelets nd ssocited unkl continuous wvelet trnsforms. More precisely, we give here generl construction llowing inverse formuls for the unkl intertwining opertor nd its dul. The contents of this pper re s follows. In Section 2, we define nd study the unkl intertwining opertor nd its dul. Section 3 is devoted to unkl wvelets nd ssocited unkl wvelet trnsforms. In the lst section, we give s ppliction of the previous results inverse formuls for the unkl intertwining opertor nd its dul. 2. The unkl intertwining opertor on nd its dul. We define nd study in this section the unkl intertwining opertor on nd its dul nd we give their properties. Nottion 2.1. We hve the following nottions: i is the spce of C -functions on with support in the intervl [,]; ii S is the spce of C -functions on, rpidly decresing together with their derivtives; iii is the subspce of consisting of functions f such tht n N, fxx n dx = ; 2.1 iv α is the subspce of consisting of functions f such tht n N, fxm n x x 2α+1 dx =, 2.2 where x, m n x = V α u n n! x; 2.3

3 UNKL WAVELETS AN APPLICATIONS TO INVESION v µ α is the mesure defined on by x 2α+1 dµ α x = dx; α+1 Γ α+1 vi L r α, 1 r +, is the spce of mesurble functions f on such tht [ 1/r f α,r = fx r dµα x] < +, 2.5 f α, = esssup fx < +. x efinition 2.2. efine the unkl intertwining opertor V α on by x k α x,yf ydy if x, x, V α fx= x f if x =, 2.6 where k α x,y = Γ α+1 x 2α 1 π Γ α+1/2 x 2 y 2 α 1/2 x +y χ] x, x [ y, 2.7 with χ ] x, x [ the chrcteristic function of the intervl ] x, x [. Theorem 2.3. i For ll f in, d x, V α f x = α fe x+ dx αi f x, 2.8 where f e resp., f is the even resp., odd prt of f, α is the iemnn-liouville integrl opertor defined in [5], nd I is the opertor given by x, I x f x = f tdt. 2.9 ii The trnsform V α is the unique topologicl isomorphism from onto itself, stisfying d V α dy f = Λ α V α f, f, 2.1 V α f = f. The inverse trnsform V 1 α is given by x, V 1 α f x = 1 d fe x+ α dx 1 α I f x, 2.11 where 1 α is the inverse opertor of α. Let f be in nd g in. The opertor t V α, defined on by the reltion V α f xgxdµ α x = fy t V α gydµ α y, 2.12

4 288 ABELLATIF JOUINI is given by y, t V α f y = k α x,yf xdµ α x, 2.13 x y where k α is the kernel given by reltion 2.7. It is clled the dul unkl intertwining opertor. It hs the following properties. Theorem 2.4. For ll f in, y, t V α f y = t d α fe y+ t α J f y, 2.14 dy where f e resp., f is the even resp., odd prt of f, t α is the Weyl integrl opertor defined in [4], nd J is the opertor given by J x f x = f ydy, x Theorem 2.5. i The trnsform t V α is topologicl isomorphism from resp., S onto itself. Moreover, f t V α f The inverse trnsform t V α 1 is given by y, t 1f V α y = t 1 d α fe y+ t 1 α J f y, 2.17 dy where t α 1 is the inverse opertor of t α. ii The trnsform t V α stisfies the trnsmuttion reltion y, t V α Λα f y = d t V α f y, f dy 3. Clssicl continuous wvelets nd unkl wvelets 3.1. Clssicl continuous wvelets on. We sy tht mesurble function g on is clssicl continuous wvelet on if it stisfies, for lmost ll x, the condition <Cg c = gλx 2 dλ < +, 3.1 λ where is the clssicl Fourier trnsform. Let ],+ [ nd let g be clssicl wvelet on in L 2. We consider the fmily g,x, x, of clssicl wvelets on in L 2 defined by g,x y = H gx y, 3.2 where H is the diltion opertor given by the reltion H f x = 1 x f. 3.3

5 UNKL WAVELETS AN APPLICATIONS TO INVESION We define, for regulr functions f on, the clssicl wvelet trnsform T g on by T g f,x = fyg,x ydy x. 3.4 This trnsform cn lso be written in the form T g f,x = f H gx, 3.5 where is the clssicl convolution product. The trnsform T g hs been studied in [5]. Severl properties re given; in prticulr, if we consider clssicl wvelet g on in L 2, we hve the following results. i Plncherel formul. For ll f in L 2, we hve fx 2 dx = 1 C c g Tg f,x 2 dx d. 3.6 ii Inversion formul. For ll f in L 1 such tht f belongs to L 1, we hve fx= 1 d Cg c T g f,yg,x ydy,.e., 3.7 where for ech x, both the inner nd the outer integrls re bsolutely convergent, but possibly not the double integrl unkl wvelets on. Using the unkl trnsform nd the unkl trnsltion opertors τ x, x, we define nd study in this section unkl wvelets on nd the unkl continuous wvelet trnsform on, nd we prove for this trnsform Plncherel nd inversion formuls. Nottion 3.1. We hve the following nottions: i σx,y,z, ρx,y,z, ndw α x,y,z re the functions defined for ll x,y,z \{} by 1 x 2 +y 2 z 2 if x,y, σx,y,z= 2xy otherwise, 1 ρx,y,z = 2 1 σx,y,z+σz,x,y+σz,y,x, W α x,y,z= 2 α+1 Γ α+1k α x, y, z ρx,y,z, where K α is the Bessel kernel; ii for ll x,y, µ α x,y is the mesure on given by W α x,y,zdµ α z if x,y, dµ x,yz α = δ x if y =, δ y if x =, where δ x is the irc mesure;

6 29 ABELLATIF JOUINI iii the unkl trnsltion opertors τ x, x, re defined on by y, τ x fy= fzdµ x,y. α 3.1 efinition 3.2. A unkl wvelet on is mesurble function g on stisfying, for lmost ll x, the condition <C g = gλx 2 dλ < +, 3.11 λ where f λ = fxψ α λ xdµ αx, λ, 3.12 nd ψ α λ z is the unkl kernel given by ψ α λ z = Γ α+1 π Γ α+1/2 1 1 e iλzt 1 t α 1/2 1+t α+1/2 dt, λ,z C Exmple 3.3. The function α t, t>, defined by x, α t = C k 4t α+1/2 e x2 /4t, 3.14 stisfies y, αt y = e ty The function gx = d/dtα t x is unkl wvelet on in, nd C g = 1/8t 2. Proposition 3.4. A function g is unkl wvelet on in resp., ifnd only if the function t V α g is clssicl wvelet on in resp.,, nd Ct V αg = C g Proof. The trnsform is topologicl isomorphism from onto itself, from α onto. We deduce then these results from Theorem 2.4. Let ],+ [ nd let g be regulr function on. We consider the function g defined by x, g x = 1 2α+1 g x It stisfies the following properties: i for g in L 2 α, the function g belongs to L 2 α nd we hve g y = gy, y ; 3.18

7 UNKL WAVELETS AN APPLICATIONS TO INVESION ii for g in resp., α, the function g belongs to resp., α nd we hve g = t V α 1oH o t V α g Let g be unkl wvelet on in L 2 α. We consider the fmily g,x,x, of unkl wvelets on in L 2 α defined by g,x y = τ x g y, y, 3.2 where τ x, x, re the unkl trnsltion opertors. Using 3.15, we deduce tht the fmily g,x, x, givenby y, g,x y = τ x d dt α t y, 3.21 is fmily of unkl wvelets on in. efinition 3.5. Let g be unkl wvelet on in L 2 α. The unkl continuous wvelet trnsform Sg on is defined for regulr functions f on by Sg f,x = fyg,x y y 2α+1 dy, >, x This trnsform cn lso be written in the form S g f,x = f g x, 3.23 where is the unkl convolution product defined by x, f gx = τ x f ygydµ α y Theorem 3.6 Plncherel formul for Sg. Let g be unkl wvelet on in L 2 α. For ll f in L 2 α, fx 2 x 2α+1 dx = 1 S C g f,x 2 x 2α+1 dx d g Proof. The function f g stisfies the reltion f g = f g.using Fubini-Tonnelli s theorem nd reltions 3.23 nd 3.18, we obtin 1 S C g f,x 2 x 2α+1 dx d g = 1 f g x 2 d x dx 2α+1 C g, = 1 f y g y 2 d y dy 2α+1 C g, = f x 2 1 gy 2 d y 2α+1 dy. C g

8 292 ABELLATIF JOUINI But from efinition 3.2, we hve for lmost ll y, 1 C g gy 2 d = 1, 3.27 then 1 C g S g f,x 2 x 2α+1 d dx = f y 2 y 2α+1 dy We then deduce the reltion The following theorem gives n inversion formul for the trnsform S g. Theorem 3.7. Let g be unkl wvelet on in L 2 α. Forf in L 1 α resp., L 2 α such tht f belongs to L 1 α resp., L 1 α L α, fx= 1 d Sg f,yg,x y y 2α+1 dy,.e., 3.29 C g where for ech x, both the inner nd the outer integrls re bsolutely convergent, but possibly not the double integrl. Proof. 199]. We obtin 3.29 by using n nlogous proof s for [5, Theorem 6.III.3, pge 4. Inversion of the unkl intertwining opertor nd of its dul by using unkl wvelets. Using the inversion formuls for the unkl continuous wvelet trnsform Sg nd clssicl continuous trnsform S g, we deduce reltions which give the inverse opertors of the unkl intertwining opertor V α nd of its dul t V α. Theorem 4.1. i Let g be unkl wvelet on in resp.,. Then for ll f in the sme spce s g, x, S g f,x = t V α 1 [ St V αg t V α f, ] x. 4.1 ii Let g be unkl wvelet on in. Then for ll f in, x, St V αgf,x = Vα 1 [ S g Vα f, ] x. 4.2 Proof. We deduce these results from reltions 2.8, 3.22, nd properties of the unkl convolution product. Theorem 4.2. Let g be unkl wvelet on in α. Then, i for ll f in, x, S g f,x = 2α V α [ S t V αg t V α f, ] x, 4.3 where is the opertor given by the reltion [ ] x, f x = 1 π x 2α+1 2 α Γ α+1 f x; 4.4

9 ii for ll f in, UNKL WAVELETS AN APPLICATIONS TO INVESION x, St V αgf,x = 2αt V α [ S g Vα f, ] x, 4.5 where is the opertor given by the reltion x, ] f x = [π 1 x 2α+1 2 α Γ α+1 f x. 4.6 Proof. We obtin these reltions from Theorem 4.1 nd the fct tht t V α g = 2α t V α g, g = 2α g. 4.7 Theorem 4.3. Let g be unkl wvelet on in α. Then, i for ll f in, x, t 1f V α x = 1 C g [ V α S t V αgf, ] d yg,x y y 2α+1 dy ; 4.8 2α+1 ii for ll f in, x, Vα 1 f x = 1 C t V k g t [ V α S d g f, ] y t V α g,x ydy α+1 Proof. We deduce 4.8 nd4.9 from Theorems 4.2, 2.4 nd reltion 2.9. emrk 4.4. We cn estblish in similr wy without mjor chnges the results given bove for the unkl intertwining opertor nd its dul in the multidimensionl cse. eferences [1] C. F. unkl, Integrl kernels with reflection group invrince, Cnd. J. Mth , no. 6, [2] A. Jouini, Continuous multiresolution nlysis ssocited with the unkl opertor on, Mth. Sci. es. J. 7 23, no. 3, [3] M. ösler, Generlized Hermite polynomils nd the het eqution for unkl opertors, Comm. Mth. Phys , no. 3, [4] E. M. Stein, Interpoltion of liner opertors, Trns. Amer. Mth. Soc , [5] K. Trimèche, The unkl intertwining opertor on spces of functions nd distributions nd integrl representtion of its dul, Integrl Trnsform. Spec. Funct , no. 4, Abdelltif Jouini: eprtment of Mthemtics, Fculty of Sciences, University of Tunis II, 16 Tunis, Tunisi E-mil ddress: bdelltif.jouini@fst.rnu.tn

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