The finite continuous Jacobi transform and its inverse
type: Item
instance of: scholarly article
title: The finite continuous Jacobi transform and its inverse
zbMATH Open document ID: 0683.44002
DOI: 10.1016/0021-9045(90)90075-2
author: Gilbert G. Walter
author: Tom H. Koornwinder
published in: Journal of Approximation Theory
publication date: 1990-01-01T00:00:00Z
full work available at URL: https://ir.cwi.nl/pub/1641
review text: The finite continuous Jacobi transform \(\hat f\) of a function f is defined by \[ \hat f(t)=\int^{\pi}_{0}f(\theta)R^{(\alpha,\beta)}_{t-\lambda}(\cos \theta)w_{\alpha,\beta}(\theta)d\theta, \] provided that the integral exists, where \(R_{\nu}^{(\alpha,\beta)}(\cos \theta)=_ 2F_ 1(- \nu,\nu +\alpha +\beta +1;\quad \alpha +1;\quad \sin^ 2(\theta /2)),\) \(\lambda =(\alpha +\beta +1)/2\) and \(w_{\alpha,\beta}(\theta)=(\sin \theta /2)^{2\alpha +1}(\cos \theta /2)^{2\beta +1}.\) The main result is an inversion formula, which is given in two forms: One contains the eigenfunctions of the Jacobi differential operator which is regular at 1 while the other involves the eigenfunctions regular at -1 as part of the inverse kernel. Finally sufficient conditions for a function to be a Jacobi transform are given.
Mathematics Subject Classification ID: 44A15
zbMATH DE Number: 4118948
zbMATH Keywords: Jacobi functions
zbMATH Keywords: finite continuous Jacobi transform
zbMATH Keywords: inversion formula
zbMATH Keywords: eigenfunctions
zbMATH Keywords: Jacobi differential operator
reviewed by: Hans-Jürgen Glaeske
MaRDI profile type: Publication
OpenAlex ID: W2143741702
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description: scientific article; zbMATH DE number 4118948