The convergence of discrete Fourier-Jacobi series

Arenas, Alberto; Ciaurri, Óscar; Labarga, Edgar

doi:10.1090/proc/14925

Cited by 1 publication

(4 citation statements)

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“…The result for the convergence of the Fourier-Jacobi series of f on the entire segment of orthogonality [−1, 1] established by Belen'kii [2] in 1989 is stated below. 3 Random Fourier-Jacobi series associated with symmetric stable process Let X(t, ω) be a symmetric stable process of index α ∈ [1,2]. Consider the random Fourier-Jacobi series…”

Section: Definition 21 ([4]mentioning

confidence: 99%

“…It is observed that the sum function of the random Fourier-Jacobi series (9) associated with symmetric stable processes X(t, ω) of index α ∈ [1,2] is weakly continuous in probability, where as the sum function of the random series (24) associated with Wiener process is continuous in quadratic mean. It is known that a function f (t, ω) is said to be weakly continuous in probability at t = t 0 , if for all > 0, lim h→0 P (|f (t 0 +h, ω)−f (t 0 , ω)| > ) = 0.…”

Section: Continuity Property Of the Sum Functionsmentioning

confidence: 99%

“…The Jacobi polynomials have great importance in a hierarchy of the orthogonal polynomial classes and found widespread use in all areas of science and engineering. Recently, Arenas, Ciaurri, and Labarga [1] studied the convergence of the Fourier-Jacobi series in a discrete version. For parameters γ, δ > −1, denote p (γ,δ) n (y), n ∈ N to be the orthonormal Jacobi polynomials with respect to the Jacobi weight ρ (γ,δ) (y) :…”

Section: Introductionmentioning

confidence: 99%

“…The literature on random Fourier series (6) gives a way to deal with random Fourier series in Jacobi polynomials. Nayak, Pattanayak, Mishra [9] discussed the convergence, summability of random Fourier series associated with symmetric stable process of index α ∈ [1,2] and shown that the random Fourier series converges in probability to a stochastic integral. We know that, if X(t, ω), t ∈ R is a continuous stochastic process with independent increments and…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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

Maharana¹,

Sahoo²

2023

Communications in Mathematics

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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.

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Section: Definition 21 ([4]mentioning

confidence: 99%

Section: Continuity Property Of the Sum Functionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%