On the summability of Random Fourier--Jacobi Series

Abstract

This article is a study on the summability of random Fourier--Jacobi series of some functions in different spaces. We consider the random series $ \sum_{n=0}^\infty a_nA_n(ω)p_n^{(γ,δ)}(y), $ where $p_n^{(γ,δ)}(y),γ,δ>-1$ are orthonormal Jacobi polynomials, the scalars $a_n$ are Fourier--Jacobi coefficients of a function $f$ and the random variables $A_n(ω)$ are Fourier--Jacobi coefficients of the symmetric stable process $X(t,ω)$ of index $α\in [1,2].$ It is established that the random Fourier--Jacobi series is $Θ$--summable in probability, if $a_n$ are the Fourier--Jacobi coefficients of function $f$ in the space $C_{[-1,1]}^{(η,τ)}.$ The Ces{á}ro $(C,ϕ),ϕ\geq1$ summability of random Fourier--Jacobi series is shown, for the symmetric stable process $X(t,ω)$ of index $α\in [1,2]$ under different conditions on the parameters $γ,δ,η$ and $τ.$ The other cases of summability, such as Riesz, Rogosinski, etc., are also discussed. Further, the N{ö}rlund summability, generalized N{ö}rlund summability, and lower triangular summability of random Fourier--Jacobi series are proved if $a_n$ are the Fourier--Jacobi coefficients of a function $f \in L_{[-1,1]}^{1,(γ,δ)},$ and $A_n(ω)$ are associated with the symmetric stable process $X(t,ω)$ of index one. It is observed that the conditions on the parameters $γ,δ$ differ from that of the conditions on $γ,δ$ for the Fourier--Jacobi series of functions $f$ in $L_{[-1,1]}^{1,(γ,δ)}.$