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OPDAM FUNCTIONS: PRODUCT FORMULA AND CONVOLUTION STRUCTURE IN DIMENSION

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OPDAM FUNCTIONS: PRODUCT FORMULA AND CONVOLUTION STRUCTURE IN DIMENSION 1 J.-PH. ANKER, F. AYADI AND M. SIFI Abstract. Let G(?,?)? (x) be the eigenfunctions of the Dunkl-Cherednik oper- ator T (?,?) on R, with ? ≥ ? ≥ ? 12 . In this paper we express the product G(?,?)? (x)G (?,?) ? (y) as an integral in terms of G (?,?) ? (z) with an explicit kernel. In general this kernel is not positive. Furthermore, by taking the so-called ratio- nal limit, we recover the product formula for the Dunkl kernels proved in [13]. We then define and study a convolution structure associated to G(?,?)? . 1. Introduction The Opdam hypergeometric functions G(?,?)? on R are normalized eigenfunctions { T (?,?)G(?,?)? (x) = i?G (?,?) ? (x) G(?,?)? (0) = 1 of the differential-difference operator T (?,?)f(x) = f ?(x) + ( (2?+ 1) cothx+ (2? + 1) tanhx )f(x)? f(?x) 2 ? ?f(?x).

  • jacobi-dunkl functions

  • kunze-stein phenomenon

  • associated convolution

  • sinh

  • jacobi functions

  • cosh z

  • called opdam- cherednik transform


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OPDAM FUNCTIONS: PRODUCT FORMULA AND CONVOLUTION STRUCTURE IN DIMENSION 1
J.-PH. ANKER, F. AYADI AND M. SIFI
Abstract.LetG(λβα)(x)be the eigenfunctions of the Dunkl-Cherednik oper-atorT(αβ)onR, withαβ≥ −2In this paper we express the product G(αβ)(x)G(λβα)(y)as an integral in terms ofG(αλβ)(z)with an explicit kernel. λ In general this kernel is not positive. Furthermore, by taking the so-called ratio-nal limit, we recover the product formula for the Dunkl kernels proved in [13]. We then define and study a convolution structure associated toG(αβλ)
1.Introduction The Opdam hypergeometric functionsG(βλα)onRare normalized eigenfunctions (TG((αβλβα))(G0(λ)αβ=)(1x) =iλG(λαβ)(x) of the differential-difference operator T(αβ)f(x) =f(x) +(2α+ 1) cothx+ (2β+ 1) tanhxf(x)2f(x)ρf(x)(1.1) Hereαβ≥ −2,α6=2 ρ=α+β+ 1andλCIn Cherednik’s notation, T(αβ)writes T(k1 k2)f(x) =f(x) +12ek12x+14ke24x(f(x)f(x))(k1+ 2k2)f(x)withα=k1+k22andβ=k22. The functionG(αβ)can be expressed as λ follows in terms of the Jacobi functionsϕ(λαβ): β) G(λαβ)(x) =ϕ(λα(x)ρ1xλϕi(αλβ)(x)(1.2) As main references we use the primary articles [11, 3] and the lecture notes [4, 12]. This paper deals with harmonic analysis for the eigenfunctionsG(βλα)We derive mainly a product formula forG(αβλ)which is analogous to the corresponding result of Flensted-Jensen and Koornwinder [6] for Jacobi functions, and of Ben Salem
Key words and phrases.Dunkl-Cherednik operator, product formula, convolution product, Opdam-Cherednik transform, Kunze-Stein phenomenon. Acknowledgements. Authors partially by the DGRST project 04/UR/15-02, the cooperation program PHC Utique/CMCU 10G 1503 and by the "Laboratoir MAPMO" (UMR 6628). 1
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J.-PH. ANKER, F. AYADI AND M. SIFI
and Ould Ahmed Salem [1] for the Jacobi-Dunkl functions. The product formula is the key information needed in order to define an associated convolution structure onRMore precisely, we deduce the product formula G(αβλ)(x)G(λαβ)(y) =ZRλC(1.3) G(λβα)(z)d(yxβα)(z)x yR from the corresponding formula forϕ(αλβ)onR+. Here(αxβy)is an explicit real valued measure with compact support onR, which may not be positive and which is uniformly bounded inx yRconclude the first part of the paper by . We recovering as a limit case the product formula for the Dunkl kernel proved in [13]. In the second part of the paper, we use the product formula (1.3) to define and study the translation operators τx(αβ)f(y) :=ZRf(z) d(yβxα)(z)
We next define the convolution product of suitable functionsfandgby fαβg(x) =ZRτx(αβ)f(y)g(y)Aαβ(|y|)dy whereAαβ(|y|) = sinh(|y|)2α+1cosh(y)2β+1We show in particular thatfαβg= gαβfand thatF(fαβg) =F(f)F(g), whereFis the so-called Opdam-Cherednik transform. Eventually we prove an analog of the Kunze-Stein phenom-enon for theαβ-convolution product ofLp-spaces. In the last part of the paper, we construct an orthonormal basis of the Hilbert spaceL2(R Aαβ(|x|)dx)generalizing the corresponding result of Koornwinder [10] forL2(R+ Aαβ(x)dx) a limit case, we recover the Hermite functions. As constructed by Rosenblum [14] inL2(R|x|2α+1dx)Our paper is organised as follows. In section 2, we recall some properties and formulas for Jacobi functions. In section 3, we give the proof of the product formula forG(λαβ)Section 4 is devoted to the translation operators and the associated convolution product. Section 5 contains a Kunze-Stein type phenomenon. In the last section 6, we construct an orthonormal basis ofL2(R Aαβ(|x|)dx).
2.Preliminaries
In this section we recall some properties of the Jacobi functions. See [6] and [7] for more details, as well as the survey [10]. Forαβ≥ −2withα6=2andλCletϕ(λβα)be the Jacobi function defined by ϕ(αβλ)(x) =2F12(1ρ+)21(ρ);α+ 1;sinh2(x)