On linear summability methods of Fourier series in polynomials orthogonal in a discrete Sobolev space
type: Item
instance of: scholarly article
title: On linear summability methods of Fourier series in polynomials orthogonal in a discrete Sobolev space
zbMATH Open document ID: 1333.42050
DOI: 10.1134/S0037446615020135
published in: Siberian Mathematical Journal
publication date: 2016-01-06T00:00:00Z
review text: The paper considers a discrete Sobolev space, i.e., the Sobolev space with the following nonstandard inner product \[ \int_{-1}^1 f(x)g(x)w(x)dx + A_1 f(1)g(1)+ B_1 f(-1)g(-1) + A_2 f'(1)g'(1) + B_2 f'( - 1)g'( - 1) \] where \(w(x)\) is an almost everywhere positive weight function on \([-1,1]\) and \(A_1\), \(B_1\), \(A_2\), \(B_2\) are nonnegative reals. The Gram-Schmidt orthogonalization generates a sequence of polynomials \(q_n(x)= q_n( x; A_1, B_1, A_2, B_2 )\) (\( n=0,1,2,\dots\); \(x \in [-1,1] \)) of power \(n\) which is orthonormal with respect to the inner product. These polynomials are in connection with various problems of mathematics and engineering. Some recent results on the convergence of Fourier series in this system of polynomials of this author are also known; this paper contains some other results concerning Toeplitz regular summation methods of Fourier series in \(q_n(x)\).
Mathematics Subject Classification ID: 42C10
Mathematics Subject Classification ID: 40C05
Mathematics Subject Classification ID: 40G05
Mathematics Subject Classification ID: 40G99
zbMATH DE Number: 6526924
zbMATH Keywords: discrete Sobolev space
zbMATH Keywords: orthogonal polynomials
zbMATH Keywords: Fourier series
zbMATH Keywords: Toeplitz regular summation methods
author: B. P. Osilenker
reviewed by: Delfina Roux
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 6526924