eng
episciences.org
Communications in Mathematics
1804-1388
2336-1298
2023-01-12
Volume 32 (2024), Issue 1
10.46298/cm.10155
10155
journal article
On the Convergence of Random Fourier-Jacobi Series of Continuous functions
Partiswari Maharana
Sabita Sahoo
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.
https://cm.episciences.org/10155/pdf
Mathematics - Functional Analysis
60G99, 40G15