No abstract

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 $\sum_{n=0}^\infty d_n r_n(\omega)\varphi_n(y)$ in orthogonal polynomials $\varphi_n(y)$ with random variables $r_n(\omega).$ The random coefficients $r_n(\omega)$ are the Fourier-Jacobi coefficients of continuous stochastic processes such as symmetric stable process and Wiener process. The $\varphi_n(y)$ 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 $\gamma,\delta$ 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.

The expected number of real zeros of a random algebraic polynomial a 0 + a 1 x + a 2 x 2 + a 3 x 3 + .... + a n−1 x n−1 depends on the types of random coefficients, with large n. In all works, the coefficients are either independent or dependent but varience of coefficients a i is one. In these cases the exepected number of real zeros is found out to be asymptotic to 2 π logn. In this article, we have considered the negatively correlated dependent random coefficients {a i } n−1 i=0 with varience σ 2i , for σ > 1 and coefficient of correlation ρ ij = −ρ |i−j| , where 0 < ρ < 1 3 . The expected number of real zeros is found to be 2 πσ logn, which is depended on σ.

Let X(t, ω), t ∈ R be a symmetric stable process with index α ∈ (1, 2] and a n be the Fourier-Jacobi coefficients of f ∈ L p , where p ≥ α.where P (γ,δ) n (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 (γ,δ) dX(t, ω) in the sense of mean and the sum function is weakly continuous in probability if the index α ∈ (1, 2] and f ∈ L p where P ≥ 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 ≥ 0, then the random Fourier-Jacobi series is (C, 1) summable in probability to the stochastic integral 1 −1 f (y, t)ρ (γ,δ) dX(t, ω).