Weighted approximation for the generalized discrete Fourier–Jacobi transform on space $L_{p}({\mathbb {T}})$

Abstract

The purpose of this work is to prove an analog of the classical Titchmarsh’s theorem (Introduction to the theory of Fourier integrals, Oxford University Press, Oxford, 1937, Theorem 84) and Younis’s Theorem (Fourier transform of Lipschitz functions on compact groups, Ph.D. Thesis, McMaster University, Hamilton, Ontario, Canada, 1974, Theorem 2.6) on the image under the discrete Fourier–Jacobi transform of a set of functions satisfying a generalized Lipschitz condition in the weighted spaces ${\mathbb {L}}_{p}([0,\pi ])$, $1<p\le 2$. For this purpose, we use a generalized translation operator which was defined by Flensted-Jensen and Koornwinder in (The convolution structure for Jacobi function expansions, Ark. Mat., 1973)