Opdam's hypergeometric functions: product formula and convolution structure in dimension 1

DOI10.1515/apam.2011.008zbMath1239.33008arXiv1004.5203OpenAlexW2964108647MaRDI QIDQ2882738

Jean-Philippe Anker, Fatma Ayadi, Mohamed Sifi

Publication date: 7 May 2012

Published in: Advances in Pure and Applied Mathematics (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1004.5203

zbMATH Keywords

convolution productDunkl-Cherednik operatorKunze-Stein phenomenonproduct formulaopdam-Cherednik transform

Mathematics Subject Classification ID

Convolution as an integral transform (44A35) Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) (33C45) Harmonic analysis on hypergroups (43A62) (L^p)-spaces and other function spaces on groups, semigroups, etc. (43A15) Hypergeometric functions associated with root systems (33C67) Orthogonal polynomials and functions associated with root systems (33C52)

Related Items (29)

Equivalence of \(K\)-functionals and modulus of smoothness constructed by Cherednik-Opdam transform ⋮ Uncertainty principles for the Heckman-Opdam transform ⋮ A new product formula involving Bessel functions ⋮ Spectral theorems associated with the Cherednik operators on \(\mathbb R^d\) ⋮ Growth Properties of the Cherednik-Opdam Transform in the Space Lp ⋮ Intertwining operators associated to a family of differential-reflection operators ⋮ A new structure of an integral operator associated with trigonometric Dunkl settings ⋮ Hausdorff operators associated with the Opdam-Cherednik transform in Lebesgue spaces ⋮ Sharp estimates for the hypergeometric functions related to root systems of type $A$ and of rank $1$ ⋮ Localization operators associated with the windowed Opdam–Cherednik transform on modulation spaces ⋮ Qualitative uncertainty principles for the windowed Opdam–Cherednik transform on weighted modulation spaces ⋮ Uncertainty principles for the windowed Opdam–Cherednik transform ⋮ Spectral theorems associated with the Jacobi-Cherednik operator ⋮ On the range of the generalized Fourier transform associated with a Cherednick type operator on the real line ⋮ An Introduction to Dunkl Theory and Its Analytic Aspects ⋮ A product formula and a convolution structure for a \(k\)-Hankel transform on \(\mathbb{R}\) ⋮ Hardy and Cowling-Price theorems associated with the Jacobi-Cherednik operator ⋮ Qualitative uncertainty principles for the Opdam–Cherednik transform ⋮ Generalized spaces of Sobolev types and applications ⋮ QUALITATIVE UNCERTAINTY PRINCIPLES FOR THE INVERSE OF THE HYPERGEOMETRIC FOURIER TRANSFORM ⋮ Некоторые оценки для обобщенного преобразования Фурье, ассоциированного с оператором Чередника - Опдама ⋮ Convolution of orbital measures on symmetric spaces: A survey ⋮ Cowling-Price's and Hardy's uncertainty principles for the generalized Fourier transform associated to a Cherednik type operator on the real line ⋮ Qualitative uncertainty principles for the hypergeometric Fourier transform ⋮ Optimal argument in the sharp Jackson inequality in the space \(L_2\) with hyperbolic weight ⋮ Characterization of the support for the hypergeometric Fourier transform of the \(W\)-invariant functions and distributions on \(\mathbb R ^d\) and Roe's theorem ⋮ On the range of the Opdam-Cherednik transform and Roe's theorem in the Cherednik setting ⋮ Some new estimates for the Cherednik-Opdam transform in the space \(L^{2}_{\alpha ,\beta}(\mathbb {R})\) ⋮ Harmonic analysis associated with the modified Cherednik type operator and quantitative uncertainty principles for its Hartley transform

Cites Work

• Convolution structure associated with the Jacobi-Dunkl operator on \(\mathbb R\)
• Jacobi functions: the addition formula and the positivity of the dual convolution structure
• Harmonic analysis for certain representations of graded Hecke algebras
• Contributions to the hypergeometric function theory of Heckman and Opdam: Sharp estimates, Schwartz space, heat kernel
• TRANSMUTATION OPERATORS AND PALEY–WIENER THEOREM ASSOCIATED WITH A CHEREDNIK TYPE OPERATOR ON THE REAL LINE
• Positivity of the Jacobi–Cherednik intertwining operator and its dual
• Jacobi Polynomials, III. An Analytic Proof of the Addition Formula
• Uniformly Bounded Representations and Harmonic Analysis of the 2 x 2 Real Unimodular Group

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