Sobolev orthogonality for the Gegenbauer polynomials \(\{ C_n^{-N+1/2}\}_{n\geq 0}\)
type: Item
instance of: scholarly article
title: Sobolev orthogonality for the Gegenbauer polynomials \(\{ C_n^{-N+1/2}\}_{n\geq 0}\)
zbMATH Open document ID: 0931.33008
DOI: 10.1016/S0377-0427(98)00124-1
author: María Álvarez de Morales
author: Teresa E. Pérez
author: Miguel A. Piñar
published in: Journal of Computational and Applied Mathematics
publication date: 2000-03-02T00:00:00Z
review text: In this paper the authors show that the Gegenbauer polynomials \(\{C_n^{N+1/2)}\}_{n\geq 0}\) for \(N=1,2,3, \dots\), are orthogonal with respect to the Sobolev inner product defined by the expression: \[ (f,g)_S= \bigl(F(1) \mid F(-1)\bigr)A \bigl(G(1) \mid G(-1)\bigr)^T+ \int^1_{-1}f^{(2N)} (x)g^{(2N)}(x) (1-x^2)^Ndx \tag{*} \] where \((F(1)\mid F(-1))= (f(1)\), \(f'(1), \dots, f^{(N-1)} (1)\), \(f(-1)\), \(f'(-1), \dots, f^{(N-1)}(-1))\), \(A\) stands for a symmetric and positive definite matrix of order \(2N\), and \(f\) and \(g\) are arbitrary polynomials. Using a certain linear operator \({\mathcal F}\) defined on the space of real polynomials, which is symmetric with respect to (*), i.e. \(({\mathcal F}f,g)_S= (f,{\mathcal F}g)_S\), they also establish several relationships between the sequences of Gegenbauer polynomials \(\{C_n^{(-N +1/2)}\}_{n\geq 0}\) and the classical ones \(\{C_n^{(N+ 1/2)}\}_{n\geq 0}\).
Mathematics Subject Classification ID: 33C45
zbMATH DE Number: 1326448
zbMATH Keywords: non-diagonal Sobolev inner product
zbMATH Keywords: Sobolev orthogonal polynomials
zbMATH Keywords: Gegenbauer polynomials
reviewed by: Nácere Hayek
MaRDI profile type: Publication
cites work: Sobolev orthogonal polynomials: The discrete-continuous case
cites work: Q4170993
cites work: Q4843473
cites work: Sobolev orthogonal polynomials and second-order differential equations
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cites work: Q4252001
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description: scientific article; zbMATH DE number 1326448