3_776_mourou.dvi EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 3, No. 6, 2010, 958-979 ISSN 1307-5543 – www.ejpam.com SPECIAL ISSUE ON COMPLEX ANALYSIS: THEORY AND APPLICATIONS DEDICATED TO PROFESSOR HARI M. SRIVASTAVA, ON THE OCCASION OF HIS 70TH BIRTHDAY Inversion of the Generalized Dunkl Intertwining Operator on R and its Dual using Generalized Wavelets W. Chabeh1, M.A. Mourou2,∗ 1 Preparatory Institute for Engineer Studies of Monastir, University of Monastir, 5019, Monastir, Tunisia 2 Department of Mathematics, College of Sciences for Girls, University of Dammam, P.O.Box 838, Dammam 31113, Saudi Arabia Abstract. We establish an inversion formula for a continuous wavelet transform associated with a class of singular differential-difference operators on R. We apply this result to derive new expressions for the inverse generalized Dunkl intertwining operator and its dual on R. 2000 Mathematics Subject Classifications: 42B20, 42C15, 44A15, 44A35 Key Words and Phrases: Differential-difference operator, Generalized Dunkl intertwining operator, Generalized continuous wavelet transform. 1. Introduction Consider the second-order singular differential operator on the real line ∆= d2 d x2 + A′(x) A(x) d d x (1) where A(x) = |x |2α+1 B(x), α > −1 2 , B being a positive C∞ even function on R. In addition we suppose that ∗Corresponding author. Email addresses: wafa_ habbah�yahoo.fr (W. Chabeh), mohamed_ali.mourou�yahoo.fr (M. Mourou) http://www.ejpam.com 958 c© 2010 EJPAM All rights reserved. W. Chabeh, M. Mourou / Eur. J. Pure Appl. Math, 3 (2010), 958-979 959 (i) A is increasing on [0,∞[ and limx→∞ A(x) =∞; (ii) A′/A is decreasing on ]0,∞[ and limx→∞ A′(x)/A(x) = 0; (iii) There exists a constant δ > 0 such that the function eδx B′(x)/B(x) is bounded for large x ∈ ]0,∞[ together with its derivatives. Lions [5] has constructed an automorphism X of the space Ee(R) of C∞ even functions on R, which intertwines ∆ and the second derivative operator d2/d x2; that is, satisfying the intertwining relation X d2 d x2 f =∆X f , f ∈ Ee(R). It is known [14] that the Lions operator X admits the integral representation X f (x) = ∫ |x | 0 G(x , y) f (y)d y, x 6= 0, where G(x , ·) is an even positive function on R, continuous on ]− |x |, |x |[ and supported in [−|x |, |x |]. Furthermore, the dual Lions operator tX f (y) = ∫ ∞ |y| G(x , y) f (x)A(x)d x , y ∈ R, is an automorphism of the space Se(R) of even Schwartz functions on R, satisfying the inter- twining relation d2 d x2 tX f = tX∆ f , f ∈ Se(R). In [8] the second author has introduced on the space E (R) of C∞ functions onR, the following operator V f =X ( fe) + d d x X I( fo), (2) where fe(x) = f (x)+ f (−x) 2 , fo(x) = f (x)− f (−x) 2 , (3) and I is the map defined by Ih(x) = ∫ x 0 h(t)d t. Mainly, he showed that V is an automorphism of E (R) satisfying for all f ∈ E (R), V d d x f = ΛV f , (4) where Λ is a first-order differential-difference operator on R given by Λ f (x) = d f d x + A′(x) A(x) � f (x)− f (−x) 2 � (5) W. Chabeh, M. Mourou / Eur. J. Pure Appl. Math, 3 (2010), 958-979 960 For A(x) = |x |2α+1, α > −1/2, the intertwining operator V reads V ( f )(x) = Γ(α+ 1)p πΓ(α+ 1/2) ∫ 1 −1 f (t x)(1− t2)α−1/2 (1+ t) d t, and referred to as the Dunkl intertwining operator of index α+ 1/2 associated with the re- flection group Z2 on R. The differential-difference operator Λ reduces to the one-dimensional Dunkl operator Dα f = d f d x + (α+ 1 2 ) f (x)− f (−x) x . Such operators have been introduced by Dunkl in connection with a generalization of the classical theory of spherical harmonics (see [1, 11] and the references therein). During the last years, the theory of Dunkl operators has found a wide area of applications in mathematics and mathematical physics. In fact, Dunkl operators have been used in the study of multivariable orthogonality structures with certain reflection symmetries [12, 16]. Moreover, they have been successfully involved in the description and solution of Calogero-Moser-Sutherland type quantum many body systems [4] . Define the dual operator t V of V on the space S (R) of Schwartz functions on R, by the relation t V f = tX ( fe) + d d x tX J( fo), (6) where J is the map defined by Jh(x) = ∫ x −∞ h(y)d y, x ∈ R. (7) In this paper, it is shown that the dual operator t V is an automorphism of S (R)which satisfies the intertwining relation d d x t V f = t VΛ f , f ∈ S (R). Moreover, the following inversion formulas for V and t V on certain specific subspaces ofS (R) are provided f = V K t V f ; f =M V t V f ; f = t VM V f ; f =K t V V f ; K andM being pseudo-differential operators. But the main contribution of this work is the determination of the inverse operators V−1 and t V−1 through a continuous wavelet transform on R associated with the differential-difference operator Λ. For examples of use of wavelet type transforms in inverse problems the reader is referred to [2, 6, 7, 10, 15] and the refer- ences therein. The content of this paper is as follows. In Section 2 we provide some harmonic W. Chabeh, M. Mourou / Eur. J. Pure Appl. Math, 3 (2010), 958-979 961 analysis results related to the differential-difference operator Λ. Next we list some basic prop- erties of the generalized Dunkl intertwining operator V and its dual t V . In section 3 we introduce the generalized continuous wavelet transform associated with Λ, and we prove for this transform Plancherel and reconstruction formulas. Using generalized wavelets, we obtain in Section 4 formulas which give the inverse operators V−1 and t V−1 on Schwartz type spaces. 2. Preliminaries In this section we provide some facts about harmonic analysis related to the differential- difference operator Λ. We cite here, as briefly as possible, only those properties actually required for the discussion. For more details we refer to [8]. Notation. We denote by - S (R) the space of C∞ functions f on R, which are rapidly decreasing together with their derivatives, i.e., such that for all m, n = 0,1, . . ., Pm,n( f ) = sup x∈R (1+ x2)m ���� dn d xn f (x) ����<∞. The topology of S (R) is defined by the semi-norms Pm,n, m, n = 0,1, . . . . - Se(R) (resp. So(R)) the subspace of S (R) consisting of even (rep. odd) functions. - B(R) the subspace of S (R) consisting of functions f such that for all n= 0,1, . . ., ∫ R f (x)bn(x)A(x)d x = 0, with bn(x) = V � yn n! � (x), V being the generalized Dunkl intertwining operator given by (2). - W (R) the subspace of S (R) consisting of functions f such that for all n= 0,1 . . ., ∫ R f (x)xnd x = 0. - H (R) the subspace of S (R) consisting of functions f such that for all n= 0,1 . . ., dn d xn f (0) = 0. Put Be(R) = Se(R)∩B(R), Bo(R) = So(R)∩B(R), We(R) = Se(R)∩W (R), Wo(R) = So(R)∩W (R), He(R) = Se(R)∩H (R), Ho(R) = So(R)∩H (R). W. Chabeh, M. Mourou / Eur. J. Pure Appl. Math, 3 (2010), 958-979 962 Remark 1. (i) Due to our assumptions on the function A there is a positive constant k such that A(x)∼ k |x |2α+1, as |x | →∞. (ii) It follows from (4) that Λbn+1 = bn (8) for all n ∈ N . Further, by [9] we have for any n ∈ N and x ∈ R, |bn(x)| ≤ k |x |n, k being a positive constant depending only on n. (iii) It is easily checked that the space S (R) is invariant under the differential-difference oper- ator Λ. For each λ ∈ C the differential-difference equation Λu = iλu, u(0) = 1, (9) admits a unique C∞ solution on R, denoted Ψλ given by Ψλ(x) = ¨ ϕλ(x)+ 1 iλ d d x ϕλ(x) if λ 6= 0, 1 if λ= 0, (10) where ϕλ designates the solution of the differential equation ∆u= −λ2u, u(0) = 1, u′(0) = 0, (11) ∆ being the differential operator defined by (1). Remark 2. (i) If A(x) = |x |2α+1, α > −1/2, then Ψλ(x) = jα(λx)+ iλx 2(α+ 1) jα+1(λx), where jγ (γ > −1/2) stands for the normalized spherical Bessel function of index γ given by jγ(z) = Γ(γ+ 1) ∞∑ n=0 (−1)n (z/2)2n n! Γ(n+ γ+ 1) (z ∈ C). W. Chabeh, M. Mourou / Eur. J. Pure Appl. Math, 3 (2010), 958-979 963 (ii) It follows by (4) and (9) that Ψλ(x) = V � eiλ·� (x) (12) for all x ∈ R and λ ∈ C. The next statement provides a new estimate for the eigenfunction Ψλ(x). Lemma 1. For all λ, x ∈ R, we have |Ψλ(x)| ≤ 1. Proof. For λ = 0, the result is obvious. For λ 6= 0, set uλ(x) = |Ψλ(x)|2 = ����ϕλ(x)+ 1 iλ d d x ϕλ(x) ���� 2 = (ϕλ(x)) 2+ 1 λ2 � d d x ϕλ(x) �2 . Notice that uλ(x) is even in x . By (11), d d x uλ(x) = 2ϕλ(x) d d x ϕλ(x)+ 2 λ2 d d x ϕλ(x) d2 d x2 ϕλ(x) = − 2 λ2 A′(x) A(x) � d d x ϕλ(x) �2 . As the function A is increasing on [0,∞[, it follows that uλ is decreasing on ]0,∞[. As uλ(0) = 1, we deduce that uλ(x)≤ 1 for all x ≥ 0. This ends the proof. Notation. For a positive Borel measure µ on R, and p = 1 or 2, we write Lp(R, dµ) for the class of measurable functions f on R for which ‖ f ‖p,µ = �∫ R | f (x)|pdµ(x) �1/p <∞. Definition 1. The generalized Fourier transform of a function f in L1(R,A(x)d x) is defined by FΛ( f )(λ) = ∫ R f (x)Ψ−λ(x)A(x)d x . (13) Remark 3. Let f ∈ L1(R,A(x)d x). By Lemma 1, it follows that FΛ( f ) is continuous on R and ||FΛ( f )||∞ ≤ ‖ f ‖1,A. An outstanding result about the generalized Fourier transform F is as follows. Theorem 1. [8] (i) For every f ∈ L1 ∩ L2(R,A(x)d x) we have the Plancherel formula ∫ R | f (x)|2A(x)d x = ∫ R |FΛ( f )(λ)|2dσ(λ). W. Chabeh, M. Mourou / Eur. J. Pure Appl. Math, 3 (2010), 958-979 964 where dσ(λ) = dλ |c(|λ|)|2 , c(z) being a continuous functions on ]0,∞[ such that c(z)−1 ∼ k1 zα+ 1 2 , as z→∞, c(z)−1 ∼ k2 zα+ 1 2 , as z→ 0, for some k1, k2 ∈ C. (ii) The generalized Fourier transform FΛ extends uniquely to a unitary isomorphism from L2(R,A(x)d x) onto L2(R, dσ). The inverse transform is given by F−1 Λ g(x) = ∫ R g(λ)Ψλ(x)dσ(λ) where the integral converges in L2(R,A(x)d x). Remark 4. (i) The tempered measure σ is called the spectral measure associated with the differential- difference operator Λ. (ii) For A(x) = |x |2α+1, α > −1/2, we have c(s) = 2α+1 Γ(α+ 1) sα+1/2 . The following lemma will play a key role in the remainder of this section. Lemma 2. The map J, given by (7), is a topological isomorphism - from So(R) onto Se(R); - fromBo(R) ontoBe(R). Proof. (i) It is sufficient to show that J maps continuously So(R) into Se(R). Let f ∈ So(R). Clearly J f is a C∞ even function on R. For n = 1,2, . . ., Pm,n(J f ) = Pm,n−1( f ). More- over, (1+ x2)m |J f (x)| ≤ (1+ x2)m ∫ ∞ |x | | f (t)|d t W. Chabeh, M. Mourou / Eur. J. Pure Appl. Math, 3 (2010), 958-979 965 ≤ ∫ ∞ |x | (1+ t2)m | f (t)|d t ≤ Pm+1,0( f ) ∫ ∞ |x | d t (1+ t2) Hence Pm,0(J f )≤ π 2 Pm+1,0( f ). (ii) Let f ∈Bo(R). By using (8) and by integrating by parts we have for any n= 0,1, . . ., ∫ R J f (x) bn(x)A(x)d x = ∫ R J f (x)Λbn+1(x)A(x)d x = − ∫ R ΛJ f (x) bn+1(x)A(x)d x = − ∫ R f (x)bn+1(x)A(x)d x = 0, which shows that J f ∈ Be(R). Conversely, let f ∈Be(R). Identity (8) together with an integration by parts yields for any n= 1,2, . . ., ∫ R f ′(x)bn(x)A(x)d x = ∫ R Λ f (x)bn(x)A(x)d x = − ∫ R f (x)Λbn(x)A(x)d x = − ∫ R f (x)bn−1(x)A(x)d x = 0, which shows that f ′ ∈Bo(R). Proposition 1. (i) For all f in S (R), we have FΛ(Λ f )(λ) = iλFΛ( f )(λ). (14) (ii) For all f in S (R), we have FΛ( f )(λ) =F∆( fe)(λ)+ iλF∆(J fo)(λ), (15) where F∆ stands for the Fourier transform related to the differential operator ∆, defined on Se(R) by F∆(h)(λ) = ∫ R h(x)ϕλ(x)A(x)d x , λ ∈ R, fe and fo being respectively the even and odd parts of f given by (3). W. Chabeh, M. Mourou / Eur. J. Pure Appl. Math, 3 (2010), 958-979 966 Proof. (i) Let f ∈ S (R). By (5), (10) and (13), FΛ(Λ f )(λ) = ∫ R � f ′o (x)+ A′(x) A(x) fo(x) � ϕλ(x)A(x)d x − 1 iλ ∫ R f ′e (x)ϕ ′ λ(x)A(x)d x = κ1 − κ2 iλ . By integrating by parts we get κ1 = ∫ R (A(x) fo(x)) ′ϕλ(x)d x = − ∫ R fo(x)ϕ ′ λ(x)A(x)d x and κ2 = ∫ R f ′e (x)ϕ ′ λ(x)A(x)d x = − ∫ R fe(x)(A(x)ϕ ′ λ(x)) ′d x = − ∫ R fe(x)∆ϕλ(x)A(x)d x = λ2 ∫ R fe(x)ϕλ(x)A(x)d x by virtue of (11). Hence κ1 − κ2 iλ = iλ ∫ R � fe(x)ϕλ(x)− fo(x) ϕ′ λ (x) iλ � A(x)d x = iλ ∫ R f (x)Φ−λ(x)A(x)d x . This clearly yields (14). (ii) If f ∈ Se(R), identity (15) is obvious. Assume f ∈ So(R). By using (10), (11), (13) and by integrating by parts we obtain FΛ( f )(λ) = − 1 iλ ∫ R f (x)ϕ′λ(x)A(x)d x = 1 iλ ∫ R J f (x)(A(x)ϕ′λ(x)) ′d x W. Chabeh, M. Mourou / Eur. J. Pure Appl. Math, 3 (2010), 958-979 967 = 1 iλ ∫ R J f (x)∆ϕλ(x)A(x)d x = iλ ∫ R J f (x)ϕλ(x)A(x)d x = iλF∆J f (λ), which completes the proof. Theorem 2. The generalized Fourier transform FΛ is a topological isomorphism - from S (R) onto itself; - fromB(R) ontoH (R). Proof. By [13] we know that the transform F∆ is a topological isomorphism - from Se(R) onto itself; - from Be(R) ontoHe(R). The result follows then from (15), Lemma 2 and the fact that the operator λ 7→ λ f is a topological isomorphism - from Se(R) onto So(R); - from He(R) ontoHo(R). Proposition 2. (i) For all f ∈ S (R), FΛ( f ) =Fu ◦ t V ( f ), (16) where Fu denotes the usual Fourier transform on R given by Fu( f )(λ) = ∫ R f (x)e−iλx d x . (ii) For all f ∈ S (R), d d x t V f = t VΛ f . (17) Proof. Assertion (i) follows by applying the usual Fourier transform Fu to both sides of (6) and by using the identity F∆h(λ) =Fu � tX h � (λ), h ∈ Se(R), (see [13]). The intertwining relation (17) follows by applying the usual Fourier transform Fu to both its sides and by using (14) and (16). W. Chabeh, M. Mourou / Eur. J. Pure Appl. Math, 3 (2010), 958-979 968 Theorem 3. The intertwining operator t V is a topological isomorphism - from S (R) onto itself; - fromB(R) ontoW (R). Proof. We deduce the result from (16), Theorem 2 and the fact that the usual Fourier transform Fu is a topological isomorphism - from S (R) onto itself; - from W (R) ontoH (R). Definition 2. (i) The generalized translation operators T x , x ∈ R, are defined on L2(R,A(x)d x) by the relation FΛ(T x f )(λ) = Ψλ(x)FΛ( f )(λ). (18) (ii) The generalized convolution product of two functions f and g in L2(R,A(x)d x) is defined by f #g(x) = ∫ R T x f (−y)g(y)A(y)d y. (19) Remark 5. Let f and g be in L2(R,A(x)d x). Then (i) By (18), Lemma 1 and Theorem 1, we deduce that T x f 2,A ≤ f 2,A (20) for any x ∈ R. (ii) It follows from (19), (20) and Schwarz inequality that f #g ∈ L∞(R) and f #g ∞ ≤ f 2,A g 2,A . (21) (iii) By virtue of (18), (19) and Theorem 1, f #g may be rewritten as f #g(x) = ∫ R FΛ( f )(λ)FΛ(g)(λ)Ψλ(x)dσ(λ). (22) Proposition 3. Let f ∈ L2(R,A(x)d x) and g ∈ L1∩L2(R,A(x)d x). Then f #g ∈ L2(R,A(x)d x), f #g 2,A ≤ f 2,A g 1,A , (23) and FΛ( f #g) =FΛ( f )FΛ(g). (24) W. Chabeh, M. Mourou / Eur. J. Pure Appl. Math, 3 (2010), 958-979 969 Proof. By Schwarz inequality, FΛ( f )FΛ(g) ∈ L1(R, dσ). Moreover, by Remark 3, FΛ( f )FΛ(g) ∈ L2(R, dσ) and FΛ( f )FΛ(g) 2,σ ≤ FΛ( f ) 2,σ g 1,A . The result follows then by combining (22) and Theorem 1. Proposition 4. If f , g ∈ S (R), then f #g ∈ S (R) and t V ( f #g) = t V f ∗ t V g, (25) where ∗ denotes the usual convolution on R. Proof. The fact that f #g ∈ S (R) follows from (24) and Theorem 2. Identity (25) follows by applying the usual Fourier transform to both its sides and by using (16) and (24). Remark 6. Notice by (24) and Theorem 2 thatB(R)#S (R)⊂B(R). 3. Generalized Wavelets Definition 3. We say that a function g ∈ L2(R,A(x)d x) is a generalized wavelet if it satisfies the admissibility condition : 0< Cg = ∫ ∞ 0 |FΛg(aλ)|2 da a <∞, (26) for almost all λ ∈ R. Remark 7. (i) The admissibility condition (26) can also be written as 0< Cg = ∫ ∞ 0 |FΛ(g)(λ)|2 dλ λ = ∫ ∞ 0 |FΛ(g)(−λ)|2 dλ λ <∞. (ii) If g is real-valued we have FΛ(g)(−λ) =FΛ(g)(λ), so (26) reduces to 0< Cg = ∫ ∞ 0 |FΛ(g)(λ)|2 dλ λ <∞. (iii) If 0 6= g ∈ L2(R,A(x)d x) is real-valued and satisfies ∃ η > 0 such that FΛ(g)(λ)−FΛ(g)(0) = O (λη), as λ→ 0+, then (26) is equivalent to FΛ(g)(0) = 0. (iv) According to (iii) and Theorem 2, each real-valued function g in B(R) is a generalized wavelet. W. Chabeh, M. Mourou / Eur. J. Pure Appl. Math, 3 (2010), 958-979 970 Proposition 5. (i) Let h ∈ L2(R, dσ) and a > 0. Then the function λ 7→ h(aλ) belongs to L2(R, dσ) and we have ‖h(a·)‖2,σ ≤ k(a)p a ‖h‖2,σ , where k(a) = sup λ>0 |c(λ)| |c(λ/a)| . (ii) For every a > 0, the dilatation operator Ha( f )(x) = 1p a f � x a � , x ∈ R, is a topological automorphism of L2(R, dσ). Proof. (i) Notice first that according to the properties of the function c(λ) given in Theorem 1, there exist two positive constants m1 and m2 such that m1 aα+1/2 ≤ k(a)≤ m2 aα+1/2 for all a > 0. We have ‖h(a·)‖22,σ = ∫ R |h(aλ)|2 dλ |c(|λ|)|2 = 1 a ∫ R |h(s)|2 |c(|s|)| 2 |c(|s|/a)|2 ds |c(|s|)|2 ≤ k2(a) a ‖h‖22,σ (ii) We deduce the result from (i). Proposition 6. Let g ∈ L2(R,A(x)d x) and a > 0. Then there exists a function ga ∈ L2(R,A(x)d x) (and only one) such that FΛ(ga)(λ) =FΛ(g)(aλ) (27) for almost every λ ∈ R. This function is given by the relation ga = 1p a F−1 Λ ◦Ha−1 ◦FΛ(g) (28) and satisfies ga 2,A ≤ k(a)p a g 2,A . W. Chabeh, M. Mourou / Eur. J. Pure Appl. Math, 3 (2010), 958-979 971 Proof. The result follows by combining Theorem 1 and Proposition 5. Remark 8. For A(x) = |x |2α+1, α > −1/2, the function ga, a > 0, is given by ga(x) = 1 a2α+2 g � x a � , x ∈ R. Proposition 7. Let g be in S (R). Then for all a > 0, the function ga belongs to S (R) and we have the relation ga = 1p a t V−1 ◦ Ha ◦ t V (g). (29) Proof. The result follows from (16), (28), Theorem 2, and the fact thatFu◦Ha = Ha−1◦Fu. Notation. For a function g in L2(R,A(x)d x) and for (a, b) ∈ ]0,∞[×R we write ga,b(x) := p a T−b ga(x), (30) where T−b are the generalized translation operators given by (18). Definition 4. Let g ∈ L2(R,A(x)d x) be a generalized wavelet. The generalized continuous wavelet transform Φg is defined for regular functions f on R by : Φg( f )(a, b) = ∫ R f (x)ga,b(x)A(x)d x . This transform can also be written in the form Φg( f )(a, b) = p a f #fga(b), (31) where # is the generalized convolution product given by (19), and ega(x) = ga(−x), x ∈ R. Lemma 3. For all f , g ∈ L2(R,A(x)d x) and all h ∈ S (R) we have the identity ∫ R f #g(x)F−1 Λ (h)(x)A(x)d x = ∫ R FΛ( f )(λ)FΛ(g)(λ)h−(λ) dσ(λ) where h−(λ) = h(−λ), λ ∈ R. Proof. Fix g ∈ L2(R,A(x)d x) and h ∈ S (R). For f ∈ L2(R,A(x)d x) put S1( f ) = ∫ R f #g(x)F−1 Λ (h)(x)A(x)d x and S2( f ) = ∫ R FΛ( f )(λ)FΛ(g)(λ)h−(λ)dσ(λ). W. Chabeh, M. Mourou / Eur. J. Pure Appl. Math, 3 (2010), 958-979 972 In view of Proposition 3 and Theorem 1, we see that S1( f ) = S2( f ) for each f ∈ L1 ∩ L2(R,A(x)d x). Moreover, by using (21), Schwarz inequality and Theorem 1 we get |S1( f )| ≤ f #g ∞ F−1 Λ (h) 1,A ≤ f 2,A g 2,A F−1 Λ (h) 1,A and |S2( f )| ≤ FΛ( f )FΛ(g) 1,σ ‖h‖∞ ≤ FΛ( f ) 2,σ FΛ(g) 2,σ ‖h‖∞ ≤ f 2,A g 2,A ‖h‖∞ , which shows that the linear functionals S1 and S2 are bounded on L2(R,A(x)d x). Therefore S1 ≡ S2, and the lemma is proved. Lemma 4. Let f1, f2 ∈ L2(R,A(x)d x). Then f1# f2 ∈ L2(R,A(x)d x) if and only if FΛ( f1)FΛ( f2) ∈ L2(R, dσ) and we have FΛ( f1# f2) =FΛ( f1)FΛ( f2) in the L2−case. Proof. Suppose f1# f2 ∈ L2(R,A(x)d x). By Lemma 3 and Theorem 1, we have for any h ∈ S (R), ∫ R FΛ( f1)(λ)FΛ( f2)(λ)h(λ) dσ(λ) = ∫ R f1# f2(x)F−1 Λ (h −)(x) A(x)d x = ∫ R f1# f2(x)F−1 Λ � h � (x)A(x)d x = ∫ R FΛ( f1# f2)(λ)h(λ) dσ(λ), which shows that FΛ( f1)FΛ( f2) = FΛ( f1# f2). Conversely, if FΛ( f1)FΛ( f2) ∈ L2(R, dσ), then by Lemma 3 and Theorem 1, we have for any h ∈ S (R), ∫ R f1# f2(x)F−1 Λ (h)(x)A(x)d x = ∫ R FΛ( f1)(λ)FΛ( f2)(λ)eh(λ) dσ(λ) = ∫ R F−1 Λ [FΛ( f1)FΛ( f2)](x)F−1 Λ (h)(x)A(x)d x , which shows, in view of Theorem 2, that f1# f2 = F−1 Λ [FΛ( f1)FΛ( f2)]. This achieves the proof. A combination of Lemma 4 and Theorem 1 gives us the following. W. Chabeh, M. Mourou / Eur. J. Pure Appl. Math, 3 (2010), 958-979 973 Lemma 5. Let f1, f2 ∈ L2(R,A(x)d x). Then ∫ R | f1# f2(x)|2A(x)d x = ∫ R |FΛ( f1)(λ)|2|FΛ( f2)(λ)|2dσ(λ) where both sides are finite or infinite. Theorem 4. Let g ∈ L2(R,A(x)d x) be a generalized wavelet. Then for all f ∈ L2(R,A(x)d x), we have the Plancherel formula ∫ R | f (x)|2A(x)d x = 1 Cg ∫ ∞ 0 ∫ R |Φg( f )(a, b)|2A(b)d b da a2 . Proof. Using (26), (27), (31), Fubini’s Theorem and Lemma 5, we have 1 Cg ∫ ∞ 0 ∫ R |Φg( f )(a, b)|2A(b)d b da a2 = = 1 Cg ∫ ∞ 0 �∫ R | f #ega(b)|2A(b)d b � da a = 1 Cg ∫ ∞ 0 �∫ R |FΛ( f )(λ)|2|FΛ(g)(aλ)|2dσ(λ) � da a = ∫ R |FΛ( f )(λ)|2 � 1 Cg ∫ ∞ 0 |FΛ(g)(aλ)|2 da a � dσ(λ) = ∫ R |FΛ( f )(λ)|2dσ(λ) The result is now a direct consequence of Theorem 1. Theorem 5. Let g ∈ L2(R,A(x)d x) be a generalized wavelet. Then for f ∈ L1 ∩ L2(R,A(x)d x) such that FΛ( f ) ∈ L1(R, dσ), we have f (x) = 1 Cg ∫ ∞ 0 �∫ R Φg( f )(a, b)ga,b(x)A(b)d b � da a2 , a.e., where, for each x ∈ R, both the inner integral and the outer integral are absolutely convergent, but possibly not the double integral. Proof. Put I (a, x) = ∫ R Φg( f )(a, b)ga,b(x)A(b)d b and J (x) = 1 Cg ∫ ∞ 0 I (a, x) da a2 . W. Chabeh, M. Mourou / Eur. J. Pure Appl. Math, 3 (2010), 958-979 974 By (30) and (31) we have I (a, x) = a ∫ R f # g̃a(b) T −x ega(b)A(b)d b. From (20), (23) and Schwarz inequality we deduce that the integral I (a, x) is absolutely convergent. On the other hand, by (18), (24) and (27), FΛ( f #ega)(λ) =FΛ( f )(λ)FΛ(g)(aλ) and FΛ(T−x g̃a)(λ) = Ψλ(−x)FΛ(g)(aλ). So using Theorem 1 we obtain I (a, x) = a ∫ R FΛ( f )(λ)Ψλ(x)|FΛ(g)(aλ)|2dσ(λ). In particular, this implies that 1 Cg ∫ ∞ 0 |I (a, x)| da a2 ≤ ∫ R |FΛ( f )(λ)| � 1 Cg ∫ ∞ 0 |FΛ(g)(aλ)|2 da a � dσ(λ) = FΛ( f ) 1,σ <∞, that is, the integral J (x) is absolutely convergent. Finally, using Fubini’s theorem we get J (x) = 1 Cg ∫ ∞ 0 �∫ R FΛ( f )(λ)|FΛ(g)(aλ)|2Ψλ(x)dσ(λ) � da a = ∫ R FΛ( f )(λ) � 1 Cg ∫ ∞ 0 |FΛ(g)(aλ)|2 da a � Ψλ(x)dσ(λ) = ∫ R FΛ( f )(λ)Ψλ(x)dσ(λ), which ends the proof in view of Theorem 1. 4. Inversion of the Intertwining Operators Using Generalized Wavelets In this section we suppose that the function |c(λ)|−2 is C∞ on ]0,∞[, and for all n ∈ N : (i) dn/dλn|c(λ)|−2 6= 0 on ]0,∞[; (ii) ∃ pn ∈ N and kn > 0 such that dn/dλn|c(λ)|−2 ≤ knλ pn for λ ≥ 1; (iii) dn/dλn|c(λ)|−2 ∼0+ anλ qn , where an ∈ R and qn ∈ Z. W. Chabeh, M. Mourou / Eur. J. Pure Appl. Math, 3 (2010), 958-979 975 Remark 9. These conditions are satisfied in the Dunkl operator case. Proposition 8. The operatorK (resp.M ) defined by K ( f ) =F−1 u � 2π |c(|λ|)|−2Fu( f ) � (32) � resp.M ( f ) =F−1 Λ � 2π |c(|λ|)|−2FΛ( f ) �� (33) is a topological automorphism of W (R) (resp. B(R)). Proof. Clearly, the mapping f 7→ 2π |c(λ)|−2 f is a topological automorphism of H (R), and its inverse is given by f 7−→ 1 2π |c(|λ|)|2 f . We deduce the result from Theorem 2 and the fact that the usual Fourier transform Fu is a topological isomorphism fromW (R) ontoH (R). Proposition 9. For f inB(R), we have M ( f ) = t V −1 ◦K ◦ t V ( f ). (34) Proof. By (16), (32) and (33), M ( f ) = F−1 Λ � 2π |c(|λ|)|−2FΛ( f ) � = t V −1 ◦F−1 u � 2π |c(|λ|)|−2Fu ◦ t V ( f ) � = t V −1 ◦K ◦ t V ( f ). Proposition 10. (i) For all f in W (R) and g in S (R), we have K ( f ∗ g) =K ( f ) ∗ g . (ii) For all f inB(R) and g in S (R), we have M ( f #g) =M ( f )#g . Proof. We have K ( f ∗ g) = F−1 u � 2π |c(|λ|)|−2Fu( f ∗ g) � = F−1 u � 2π |c(|λ|)|−2Fu( f )Fu(g) � = ¦ F−1 u � 2π |c(|λ|)|−2Fu( f ) �© ∗ g = K ( f ) ∗ g and M ( f #g) = F−1 Λ � 2π |c(|λ|)|−2FΛ( f #g) � W. Chabeh, M. Mourou / Eur. J. Pure Appl. Math, 3 (2010), 958-979 976 = F−1 Λ � 2π |c(|λ|)|−2FΛ( f )FΛ(g) � = ¦ F−1 Λ � 2π |c(|λ|)|−2FΛ( f ) �© #g = M ( f )#g, which ends the proof. Theorem 6. 1. The intertwining operator V is a topological isomorphism from W (R) onto B(R). 2. We have the following inverse formulas for V and t V : (a) For f ∈B(R), f = V K t V ( f ); (35) f =M V t V ( f ). (36) (b) For f ∈W (R), f =K t V V ( f ); (37) f = t VM V ( f ). (38) Proof. Let f ∈B(R). From (12), (16) and Theorem 1 we have for all x ∈ R, f (x) = ∫ R FΛ( f )(λ)Ψλ(x) dλ |c(|λ|)|2 = V �∫ R FΛ( f )(λ) eiλ· dλ |c(|λ|)|2 � (x) = V � 1 2π ∫ R � 2π |c(|λ|)|−2Fu ◦ t V ( f ) � eiλ·dλ � (x) = VK t V ( f )(x). This when combined with (34) yields formula (36). By replacing f respectively by V f and t V−1 f into formulas (35) and (36), we regain identities (37) and (38). From (35), Propo- sition 8 and Theorem 3, we deduce that V is a topological isomorphism from W (R) onto B(R). In order to invert the intertwining operators V and t V we shall need some technical lemmas. Lemma 6. For all f in W (R) and g in S (R), we have V ( f ∗ g ) = V ( f )# t V −1 (g). (39) Proof. By using relations (25), (35), (37) and Proposition 10(i) we have V−1 � V ( f )# t V−1(g) � = K t V � V ( f )# t V−1(g) � = K � t V V ( f ) ∗ g � = �K t V V ( f ) � ∗ g = f ∗ g. W. Chabeh, M. Mourou / Eur. J. Pure Appl. Math, 3 (2010), 958-979 977 Definition 5. The classical continuous wavelet transform on R is defined for regular functions by Sg( f )(a, b) = ∫ R f (x) g0 a,b (x)d x , a > 0, b ∈ R, where g0 a,b (x) := 1p a g � x − b a � The function g is a classical wavelet on R, i.e., a function in L2(R, d x) satisfying the admissibility condition: 0< C0 g = ∫ ∞ 0 |Fu(g)(aλ)|2 da a <∞, for almost all λ ∈ R. A more complete and detailed discussion of the properties of the classical wavelet trans- form on R can be found in [3], from which we have the following inversion formula. Theorem 7. Let g ∈ L2(R, d x) be a classical wavelet. If both f and Fu( f ) are in L1(R, d x) then we have f (x) = 1 C0 g ∫ ∞ 0 �∫ R Sg( f )(a, b)g0 a,b(x)d b � da a2 for almost every x ∈ R. Remark 10. According to (16) and Definitions 3, 5, g ∈ S (R) is a generalized wavelet, if and only if, t V (g) is a classical wavelet and we have: C0 t V(g) = Cg . (40) Lemma 7. Let g ∈W (R) be real-valued. Then for all f ∈ S (R) we have ΦVK g( f )(a, b) =MV � Sg � t V f � (a, ·) � (b). Proof. Notice that VK g = t V−1 g by virtue of (35). Further, g is a classical wavelet according to [3]. So it follows from Remark 10 that VK g ∈ B(R) is a generalized wavelet and CVK g = C0 g . (41) Due to (25), (29), (31), (35), (38) and Definition 5 we have ΦVK g( f )(a, b) = p a f #(VK g)ea(b) = p a t V−1 � t V f ∗ t V (VK g)ea � (b) = t V−1 � t V f ∗Ha � t V VK eg�� (b) = MV � t V f ∗Ha(eg) � (b) = MV � Sg � t V f � (a, ·) � (b). REFERENCES 978 Lemma 8. Let g ∈B(R) be real-valued. Then for all f ∈W (R), we have S t V g( f )(a, b) =K t V � Φg(V f )(a, ·) � (b). Proof. Observe that by Remarks 7(iv) and 10, t V g is a classical wavelet. Using (29), (31), (39) and Definition 5 we have V � S t V g( f )(a, ·) � (b) = V � f ∗ Ha � t V eg �� (b) = p a V � f ∗ t V (ega) � (b) = p a V ( f )# ega(b) = Φg(V f )(a, b). Thus S t V g( f )(a, b) = V−1 [Φg(V f )(a, ·)](b) = K t V[Φg(V f )(a, ·)](b) by virtue of (35). We can now state our main result. Theorem 8. (i) Let g ∈W (R) be real-valued. Then for all f ∈ S (R) we have t V −1 f (x) = 1 C0 g ∫ ∞ 0 �∫ R MV [Sg( f )(a, ·)](b) (VK g)a,b(x)A(b)d b � da a2 . (ii) Let g ∈B(R) be real-valued. Then for all f ∈B(R) we have V−1 f (x) = 1 Cg ∫ ∞ 0 �∫ R K t V[Φg( f )(a, ·)](b)� t V g �0 a,b (x)d b � da a2 . Proof. The result follows by combining Theorems 5, 7, Lemmas 7, 8 and identities (40), (41). References [1] MFE De Jeu. The Dunkl transform. Invent. Math, 133:147-162, 1993. [2] M Holschneider. Inverse Radon transform through inverse wavelet transform. Inverse Problems, 7:853-861, 1991. REFERENCES 979 [3] TH Koornwinder. The continuous wavelet transform. Wavelets : An Elementary Treate- ment of Theory and Applications. Edited by T.H. Koornwinder , World Scientific , pages 27-48, 1993. [4] L Lapointe and L Vinet. Exact operator solution of the Calogero-Sutherland model. Comm. Math. Phys., 178:425-452, 1996. [5] JL Lions. Equations différentielles opérationnelles et problèmes aux limites. Springer- Verlag, Berlin, 1961. [6] MA Mourou and K Trimèche. Inversion of the Weyl integral transform and the Radon transform on Rn using generalized wavelets. Monatshefte für Mathematik, 126:73-83, 1998. [7] MA Mourou and K Trimèche. Calderon’s formula associated with a differential operator on (0,∞) and inversion of the generalized Abel transform. Journal of Fourier Analysis and Applications, 4:229-245, 1998. [8] MA Mourou and K Trimèche. Transmutation operators and Paley-Wiener theorem as- sociated with a singular Differential-Difference operator on the real line. Analysis and Applications, 1:43-69, 2003. [9] MA Mourou. Taylor series associated with a differential-difference operator on the real line. Journal of Computational and Applied Mathematics, 153:343-354, 2003. [10] MA Mourou. Inversion of the dual Dunkl-Sonine integral transform on R using Dunkl wavelets. SIGMA, 5:1-12, 2009. [11] M Rösler. Positivity of Dunkl’s intertwining operator. Duke Math. J., 98:445-463, 1999. [12] M Rösler. Generalized Hermite polynomials and the heat equation for Dunkl operators. Comm. Math. Phys., 192:519-542, 1998. [13] K Trimèche. Inversion of the Lions transmutation operators using generalized wavelets. Appl. and comput. Harm. Anal., 4:97-112, 1997. [14] K Trimèche. Transformation intégrale de Weyl et théorème de Paley-Wiener associés àun opérateur différentiel singulier sur [0,+∞[. J. Math. Pures Appl., 60:51-98, 1981. [15] K Trimèche. Generalized wavelets and hypergroups. Gordon and Breach Publishing group, 1997. [16] Y Xu. Intertwining operator and h-harmonics associated with reflection groups. Canad. J. Math., 50:193-209, 1998.