arXiv subjects

Partiswari Maharana

Publications and source records attributed to Partiswari Maharana.

4 recordsLinked to original sources

arXiv2023

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

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 $\int_{-1}^1f(y, t)ρ^{(γ,δ)}dX(t,ω).$

math.PR↗

arXiv2023

On the Convergence of Random Fourier-Jacobi Series of Continuous functions

The interest in orthogonal polynomials and random Fourier series in numerous branches of science and a few studies on random Fourier series in orthogonal polynomials inspired us to focus on random Fourier series in Jacobi polynomials. In the present note, an attempt has been made to investigate the stochastic convergence of some random Jacobi series. We looked into the random series $\sum_{n=0}^\infty d_n r_n(ω)φ_n(y)$ in orthogonal polynomials $φ_n(y)$ with random variables $r_n(ω).$ The random coefficients $r_n(ω)$ are the Fourier-Jacobi coefficients of continuous stochastic processes such as symmetric stable process and Wiener process. The $φ_n(y)$ are chosen to be the Jacobi polynomials and their variants depending on the random variables associated with the kind of stochastic process. The convergence of random series is established for different parameters $γ,δ$ of the Jacobi polynomials with corresponding choice of the scalars $d_n$ which are Fourier-Jacobi coefficients of a suitable class of continuous functions. The sum functions of the random Fourier-Jacobi series associated with continuous stochastic processes are observed to be the stochastic integrals. The continuity properties of the sum functions are also discussed.

math.FA↗

arXiv2022

On the Convergence of Random Fourier--Jacobi Series in $L_{[-1,1]}^{p,(η,τ)}$ space

Liu and Liu introduced the random Fourier transform, which is a random Fourier series in Hermite functions, and applied it to image encryption and decryption. They expected its applications in optics and information technology. These motivated us to look into random Fourier series in orthogonal polynomials. Recently, we have established the convergence of random Fourier--Jacobi series $\sum\limits_{n=0}^\infty d_n r_n(ω)φ_n(y),$ where $φ_n(y)$ are the orthonormal Jacobi polynomials $p_n^{(γ,δ)}(y),$ $r_n(ω)$ are random variables associated with stochastic processes like the Wiener process, the symmetric stable process, and the scalars $d_n$ are the Fourier--Jacobi coefficients of functions in some classes of continuous functions. It is observed that the mode of convergence of the random series depends on the choice of the scalars $d_n$ and the stochastic processes. In this article, we have investigated the scalars $d_n,$ which are chosen to be the Fourier--Jacobi coefficients of functions in some weighted $L_{[-1,1]}^{p,(η,τ)}$ spaces, so that the random series converges. Further, the continuity property of the sum functions is studied.

math.FA↗

arXiv2019

Real zeros of algebraic polynomials with nonidentical dependent random coefficients

The expected number of real zeros of an algebraic polynomial $a_0+a_1x+a_2x^2+a_3x^3+....+a_{n-1}x^{n-1}$ depends on the types of random coefficients, with large $n.$ In this article, we show that when the random coefficients $\{a_i\}_{i=1}^{n-1}$ are assumed to be negatively dependent with $var(a_i)=σ^{2i}$ and correlation between any two coefficients for $i\neq j,$ assumed to be $ρ_{ij}=-ρ^{|i-j|},$ where $0<ρ<\frac{1}{3}$, then the expected number of real zeros is asymptotically equal to $\frac{2}{πσ}logn.$

math.FA↗