On the summability of random Fourier–Jacobi series

This article explores the summability of random Fourier-Jacobi seriesin various functions spaces, where q (γ,δ) i (y), γ, δ > -1 are the orthonormal Jacobi polynomials and bi are scalars.The random coefficients Bi(ϖ) are the Fourier-Jacobi coefficients of a symmetric stable process Z(t, ϖ) of index χ ∈ [1, 2].It is established that the random Fourier series in Jacobi polynomials is summable in Υ means, Cesàro means, Rogosinski means, etc., in the sense of probability if bi are the Fourier-Jacobi coefficients of a function f ∈ C (η,τ ) [-1,1] space, under different conditions on the Jacobi parameters γ, δ, η and τ.Further, the Nörlund summability, generalized Nörlund summability and lower triangular summability of random Jacobi series associated with the symmetric stable process Z(t, ϖ) of index one is established for functions belonging to the L 1,(γ,δ) [-1,1] space.It has been noted that the conditions on the Jacobi parameters for random series differ from that of the Fourier-Jacobi series.