On the Convergence of Random Fourier-Jacobi Series of Continuous functions

TL;DR

This paper investigates the stochastic convergence of random Fourier series in Jacobi polynomials, focusing on series with coefficients from continuous stochastic processes like Wiener and stable processes, and explores their sum functions and continuity.

Contribution

It establishes convergence conditions for random Jacobi series with coefficients from continuous stochastic processes, extending understanding of their stochastic integral representations.

Findings

Convergence of random Jacobi series depends on parameters and coefficients.

Sum functions of the series are related to stochastic integrals.

Continuity properties of the sum functions are analyzed.

Abstract

The interest in orthogonal polynomials and random Fourier series in numerous branches of science and a few studies on random Fourier series in orthogonal polynomials inspired us to focus on random Fourier series in Jacobi polynomials. In the present note, an attempt has been made to investigate the stochastic convergence of some random Jacobi series. We looked into the random series $\sum_{n=0}^\infty d_n r_n(\omega)\varphi_n(y)$ in orthogonal polynomials $\varphi_n(y)$ with random variables $r_n(\omega).$ The random coefficients $r_n(\omega)$ are the Fourier-Jacobi coefficients of continuous stochastic processes such as symmetric stable process and Wiener process. The $\varphi_n(y)$ are chosen to be the Jacobi polynomials and their variants depending on the random variables associated with the kind of stochastic process. The convergence of random series is established for different parameters $\gamma,\delta$ of the Jacobi polynomials with corresponding choice of the scalars $d_n$ which are Fourier-Jacobi coefficients of a suitable class of continuous functions. The sum functions of the random Fourier-Jacobi series associated with continuous stochastic processes are observed to be the stochastic integrals. The continuity properties of the sum functions are also discussed.