On the summability of Random Fourier--Jacobi Series

TL;DR

This paper investigates the summability properties of random Fourier--Jacobi series involving symmetric stable processes, establishing various summability results and conditions for functions in different function spaces.

Contribution

It introduces new summability results for random Fourier--Jacobi series, including $ heta$-summability, Cesàro, Riesz, Rogosinski, and N{"o}rlund methods, under diverse parameter conditions.

Findings

Random Fourier--Jacobi series are $ heta$-summable in probability.

Cesàro $(C, ext{phi})$ summability is established under specific parameter conditions.

N{"o}rlund and lower triangular summability are proved for functions in $L^{1,( ext{gamma}, ext{delta})}$.

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(\omega)p_n^{(\gamma,\delta)}(y), $ where $p_n^{(\gamma,\delta)}(y),\gamma,\delta>-1$ are orthonormal Jacobi polynomials, the scalars $a_n$ are Fourier--Jacobi coefficients of a function $f$ and the random variables $A_n(\omega)$ are Fourier--Jacobi coefficients of the symmetric stable process $X(t,\omega)$ of index $\alpha \in [1,2].$ It is established that the random Fourier--Jacobi series is $\Theta$--summable in probability, if $a_n$ are the Fourier--Jacobi coefficients of function $f$ in the space $C_{[-1,1]}^{(\eta,\tau)}.$ The Ces{\'a}ro $(C,\phi),\phi \geq1$ summability of random Fourier--Jacobi series is shown, for the symmetric stable process $X(t,\omega)$ of index $\alpha \in [1,2]$ under different conditions on the parameters $\gamma,\delta,\eta$ and $\tau.$ The other cases of summability, such as Riesz, Rogosinski, etc., are also discussed. Further, the N{\"o}rlund summability, generalized N{\"o}rlund summability, and lower triangular summability of random Fourier--Jacobi series are proved if $a_n$ are the Fourier--Jacobi coefficients of a function $f \in L_{[-1,1]}^{1,(\gamma,\delta)},$ and $A_n(\omega)$ are associated with the symmetric stable process $X(t,\omega)$ of index one. It is observed that the conditions on the parameters $\gamma,\delta$ differ from that of the conditions on $\gamma,\delta$ for the Fourier--Jacobi series of functions $f$ in $L_{[-1,1]}^{1,(\gamma,\delta)}.$