On the Convergence of Random Fourier--Jacobi Series in $L_{[-1,1]}^{p,(\eta,\tau)}$ space

TL;DR

This paper studies the convergence of random Fourier--Jacobi series in weighted $L^p$ spaces, analyzing how the choice of coefficients and stochastic processes affect convergence and continuity of the series sum.

Contribution

It establishes convergence conditions for random Fourier--Jacobi series with coefficients from weighted $L^p$ spaces and examines the continuity of the resulting sum functions.

Findings

Convergence depends on the choice of Fourier--Jacobi coefficients and stochastic processes.

The series converges in the weighted $L^p$ space under certain conditions.

Continuity of the sum functions is established under specific settings.

Abstract

Liu and Liu introduced the random Fourier transform, which is a random Fourier series in Hermite functions, and applied it to image encryption and decryption. They expected its applications in optics and information technology. These motivated us to look into random Fourier series in orthogonal polynomials. Recently, we have established the convergence of random Fourier--Jacobi series $\sum\limits_{n=0}^\infty d_n r_n(\omega)\varphi_n(y),$ where $\varphi_n(y)$ are the orthonormal Jacobi polynomials $p_n^{(\gamma,\delta)}(y),$ $r_n(\omega)$ are random variables associated with stochastic processes like the Wiener process, the symmetric stable process, and the scalars $d_n$ are the Fourier--Jacobi coefficients of functions in some classes of continuous functions. It is observed that the mode of convergence of the random series depends on the choice of the scalars $d_n$ and the stochastic processes. In this article, we have investigated the scalars $d_n,$ which are chosen to be the Fourier--Jacobi coefficients of functions in some weighted $L_{[-1,1]}^{p,(\eta,\tau )}$ spaces, so that the random series converges. Further, the continuity property of the sum functions is studied.