On Summability of Random Fourier-Jacobi Series associated with Stable Process

Abstract

Let $X(t,\omega ),$ $t\in \text{R}$ be a symmetric stable process with index $\alpha \in (1,2]$ and $a_n$ be the Fourier-Jacobi coefficients of $f\in L^p,$ where $p\ge \alpha .$ For $\gamma ,\delta >0,$ $t\in [-1,1],$ define $A_n(\omega )={\int }_{-1}^1{P}_n^{(\gamma ,\delta )}(t){\rho }^{(\gamma ,\delta )}dX(t,\omega )$ where $P_n^{(\gamma ,\delta )}(t)$ are orthogonal Jacobi polynomials. The $A_n(\omega )$ exists in the sense of mean. In this paper, it is shown that the random Fourier-Jacobi series ${\sum }_{n=0}^{\infty }{a}_n{A}_n(\omega )P_n^{(\gamma ,\delta )}(y)$ converges to the stochastic integral ${\int }_{-1}^{1}f(y,t){\rho }^{(\gamma ,\delta )}dX(t,\omega )$ in the sense of mean and the sum function is weakly continuous in probability if the index $\alpha \in (1,2]$ and $f\in L^p$ where $P\ge \alpha .$ However, it is shown that if the index $\alpha$ is one and $f$ is in the weighted space of continuous function $C^{(\eta ,\tau )}(-1,1),$ for $\eta ,\tau \ge 0,$ then the random Fourier-Jacobi series is $(C,1)$ summable in probability to the stochastic integral ${\int }_{-1}^{1}f(y,t){\rho }^{(\gamma ,\delta )}dX(t,\omega ).$

arXiv commentThis paper is divided into two paper

References in corpus (1)

• On a power series involving classical orthogonal polynomials