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| HAL: hal-00476400, version 2 |
| arXiv: 1004.5203 |
| Detailed view |  Export this paper |
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| Adv. Pure Appl. Math. (2011) 27 pp. |
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| Available versions: | v1 (2010-04-29) | v2 (2011-01-06) | v3 (2011-05-18) |
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| Opdam's hypergeometric functions: product formula and convolution structure in dimension 1 |
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| Jean-Philippe Anker 1Fatma Ayadi 1 |
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| (2011) |
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| Let $G_{\lambda}^{(\alpha,\beta)}$ be the eigenfunctions of the Dunkl-Cherednik operator $T^{(\alpha,\beta)}$ on $\mathbb{R}$. In this paper we express the product $G_{\lambda}^{(\alpha,\beta)}(x)G_{\lambda}^{(\alpha,\beta)}(y)$ as an integral in terms of $G_{\lambda}^{(\alpha,\beta)}(z)$ with an explicit kernel. In general this kernel is not positive. Furthermore, by taking the so-called rational limit, we recover the product formula of M. Rösler for the Dunkl kernel. We then define and study a convolution structure associated to $G_{\lambda}^{(\alpha,\beta)}$. |
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| 1: | Mathématiques - Analyse, Probabilités, Modélisation - Orléans (MAPMO) |
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Université d'Orléans – CNRS : UMR7349 |
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| 2: | Analyse Mathématique et Applications |
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Ecole Préparatoire aux Etudes d'Ingénieurs de Tunis – Université Tunis El Manar |
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| Subject | : | Mathematics/Classical Analysis and ODEs Mathematics/Functional Analysis |
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| Dunkl-Cherednik operator – Opdam-Cherednik transform – product formula – convolution product – Kunze-Stein phenomenon |
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| hal-00476400, version 2 | |
| http://hal.archives-ouvertes.fr/hal-00476400 | |
| oai:hal.archives-ouvertes.fr:hal-00476400 | |
| From: Fatma Ayadi | |
| Submitted on: Wednesday, 5 January 2011 21:40:12 | |
| Updated on: Tuesday, 19 April 2011 11:11:55 | |
