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

Abstract

Let $X(t,\omega),$ $t \in \textit{R}$ 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 $p \geq \alpha.$ 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 $\int_{-1}^1f(y,t)\rho^{(\gamma,\delta)}dX(t,\omega)$ 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 $P \geq \alpha.$ 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 $\eta, \tau \geq 0,$ then the random Fourier-Jacobi series is $(C,1)$ summable in probability to the stochastic integral $\int_{-1}^1f(y, t)\rho^{(\gamma,\delta)}dX(t,\omega).$