Partiswari Maharana Sabita Sahoo

arXiv2024

On the Convergence of Random Fourier--Jacobi Series in weighted $\mathrm{L}_{[-1,1]}^{\mathrm{p},(ζ,η)}$ Space

In the present paper, the random series $\sum\limits_{m=0}^\infty c_m C_m(\varpi)q_m^{(ζ,η)}(u)$ in orthogonal Jacobi polynomials $q_m^{(ζ,η)}(u)$ is discussed. The scalars $c_m$ are Fourier--Jacobi coefficients of a function in the weighted space $\mathrm{L}_{[-1,1]}^\mathrm{p}(dμ_{ζ,η}),\mathrm{p}>1.$ The random variables $C_m(\varpi)$ are chosen to be the Fourier--Jacobi coefficients of symmetric stable process $Y_{ζ,η}(v,\varpi)$ of index $χ\in [1,2]$ for $ζ,η\geq 0,$ which are not independent. We prove that, under certain conditions on $\mathrm{p},ζ$ and $η,$ the random Fourier--Jacobi series converges in probability to the stochastic integral \begin{equation*} \int_{-1}^1 \mathfrak{g}(u,v)dY_{ζ,η}(v,\varpi). \end{equation*} We also establish the existence of this integral in the sense of probability for $\mathfrak{g} \in \mathrm{L}_{[-1,1]}^\mathrm{p}(dμ_{ζ,η}).$

math.FA↗

arXiv2023

On the summability of Random Fourier--Jacobi Series

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 $L_{[-1,1]}^{1,(γ,δ)}.$

math.FA↗