episciences.org_10155_20230401152259257
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Communications in Mathematics
23361298
01
12
2023
Volume 32 (2024), Issue 1
On the Convergence of Random FourierJacobi 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 FourierJacobi 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 FourierJacobi coefficients of a suitable class
of continuous functions. The sum functions of the random FourierJacobi series
associated with continuous stochastic processes are observed to be the
stochastic integrals. The continuity properties of the sum functions are also
discussed.
01
12
2023
10155
https://arxiv.org/licenses/nonexclusivedistrib/1.0
arXiv:2210.06655
10.48550/arXiv.2210.06655
https://arxiv.org/abs/2210.06655v1
10.46298/cm.10155
https://cm.episciences.org/10155

https://cm.episciences.org/10412/pdf

https://cm.episciences.org/10412/pdf