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

Abstract

Let $X(t,ω),$ $t \in \textit{R}$ be a symmetric stable process with index $α\in (1,2]$ and $a_n$ be the Fourier-Jacobi coefficients of $f \in L^p,$ where $p \geq α.$ For $γ, δ> 0,$ $t \in [-1,1],$ define $A_n(ω)=\int_{-1}^1 P_n^{(γ,δ)}(t)ρ^{(γ,δ)}dX(t,ω)$ where $P_n^{(γ,δ)}(t)$ are orthogonal Jacobi polynomials. The $A_n(ω)$ exists in the sense of mean. In this paper, it is shown that the random Fourier-Jacobi series $\sum_{n=0}^\infty a_n A_n(ω)P_n^{(γ,δ)}(y)$ converges to the stochastic integral $\int_{-1}^1f(y,t)ρ^{(γ,δ)}dX(t,ω)$ in the sense of mean and the sum function is weakly continuous in probability if the index $α\in (1,2]$ and $f \in L^p$ where $P \geq α.$ However, it is shown that if the index $α$ is one and $f$ is in the weighted space of continuous function $C^{(η, τ)}(-1,1),$ for $η, τ\geq 0,$ then the random Fourier-Jacobi series is $(C,1)$ summable in probability to the stochastic integral $\int_{-1}^1f(y, t)ρ^{(γ,δ)}dX(t,ω).$