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

Let $X(t,\omega ),$ $t\in \mathit{\text{R}}$ 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$p \geq \alpha.$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$P_n^{(\gamma,\delta)}(t)$areorthogonalJacobipolynomials\mathrm{.}The$A_n(\omega)$ exists in the sense of mean. In this paper, it is shown that the random Fourier-Jacobi series $\sum_{n=0}^\infty a_n A_n(\omega)P_n^{(\gamma,\delta)}(y)$convergestothestochasticintegral$\int_{-1}^1f(y,t)\rho^{(\gamma,\delta)}dX(t,\omega)$ 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 $\eta ,\tau \ge 0,$ then the random Fourier-Jacobi series is $(C,1)$ summable in probability to the stochastic integral ${\int }_{-1}^{1}f(y,t){\rho }^{(\gamma ,\delta )}dX(t,\omega )\mathrm{.}$