The finite continuous Jacobi transform and its inverse
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.
scientific article; zbMATH DE number 4118948