Harmonic analysis associated with the Jacobi-Dunkl operator on \(]-\frac{\pi}{2},\frac{\pi}{2}[\)
type: Item
instance of: scholarly article
title: Harmonic analysis associated with the Jacobi-Dunkl operator on \(]-\frac{\pi}{2},\frac{\pi}{2}[\)
zbMATH Open document ID: 1062.43011
author: Frej Chouchene
published in: Journal of Computational and Applied Mathematics
publication date: 2005-04-18T00:00:00Z
review text: In this paper the author considers the Jacobi-Dunkl operator \(\Lambda_{\alpha,\beta}\) on \(]-{\pi\over 2},{\pi\over 2}[\) given by \[ \Lambda_{\alpha,beta} f(\theta)= {d\over d\theta} f(\theta)+ {A_{\alpha,\beta}'(x)\over A_{\alpha,\beta}(x)} \Biggl({f(\theta)- f(-\theta)\over 2}\Biggr), \] where \(A_{\alpha,\beta}(\theta)= 2^{2\rho}(\sin|\theta|)^{2\alpha+ 1}(\cos\theta)^{2\beta+1}\), \(\alpha\geq \beta>-{1\over 2}\), \(\alpha\neq -{1\over 2}\), \(\rho= \alpha+ \beta+ 1\). The eigenfunctions of this operator equal to 1 at zero are called Jacobi-Dunkl polynomials. He proves for these polynomials a Laplace integral representation, and with the aid of these polynomials he defines the Jacobi-Dunkl transform and he establishes for this transform an inversion formula and a Parseval formula and a Paley-Wiener theorem.
reviewed by: Khalifa Trimèche
Mathematics Subject Classification ID: 43A32
Mathematics Subject Classification ID: 42A38
zbMATH DE Number: 2157660
zbMATH Keywords: Jacobi-Dunkl operator
zbMATH Keywords: Laplace integral representations
zbMATH Keywords: Harmonic analysis
MaRDI profile type: Publication
full work available at URL: j.cam.2004.02.025
OpenAlex ID: W1511432181
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DOI: 10.1016/J.CAM.2004.02.025
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description: scientific article; zbMATH DE number 2157660