与稳定过程相关的随机Fourier-Jacobi级数的可和性
On Summability of Random Fourier-Jacobi Series associated with Stable Process
摘要：
设$X(t,ω),$ $t \in \textit{R}$是一个具有指标$α\in (1,2]$的对称稳定过程，$a_n$是$f \in L^p,$的Fourier-Jacobi系数，其中$p \geq α.$对于$γ, δ> 0,$ $t \in [-1,1],$定义$A_n(ω)=\int_{-1}^1 P_n^{(γ,δ)}(t)ρ^{(γ,δ)}dX(t,ω)$，其中$P_n^{(γ,δ)}(t)$是正交Jacobi多项式。$A_n(ω)$是在平均值的意义上存在的。本文证明了随机Fourier-Jacobi级数$\sum_{n=0}^\infty a_n A_n(ω)P_n^{(γ,δ)}(y)$在均值意义下收敛于随机积分$\int_{-1}^1f(y,t)ρ^{(γ,δ)}dX(t,ω)$，且其和函数在概率意义下弱连续，如果指标$α\in (1,2]$和$f \in L^p$满足其中$P \geq α.$，证明了若指标$α$为1且$f$在$η, τ\geq 0,$的连续函数$C^{(η, τ)}(-1,1),$的加权空间中，则随机Fourier-Jacobi级数对随机积分$\int_{-1}^1f(y, t)ρ^{(γ,δ)}dX(t,ω).$依概率$(C,1)$可和
英文摘要：
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 $\int_{-1}^1f(y, t)ρ^{(γ,δ)}dX(t,ω).$
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