A family of Sobolev orthogonal polynomials on the unit circle
type: Item
instance of: scholarly article
title: A family of Sobolev orthogonal polynomials on the unit circle
zbMATH Open document ID: 0952.42013
DOI: 10.1016/S0377-0427(99)00040-0
published in: Journal of Computational and Applied Mathematics
publication date: 2000-10-26T00:00:00Z
review text: The authors consider the sequence of monic polynomials \(\Phi_n\) orthogonal with respect to the inner product \[ \langle f,g \rangle_s=\int_0^{2\pi} f(e^{i\theta}) \overline{g(e^{i\theta})} \frac{d \theta}{|e^{i \theta}-\alpha|^2} + \frac{1}{\lambda} \int_0^{2\pi} f'(e^{i\theta}) \overline{g'(e^{i\theta})} d \theta,, \] where \(\lambda >0\) and \(|\alpha|<1\). Algebraic expressions for \(\Phi_n\) and for their norms, and a three-term recurrence for \(\Phi_n\) are derived. From here, asymptotics for \(\Phi_n\) and the behavior of their zeros as \(n \to \infty\) is established. The paper is very much connected with another one by the same authors [\textit{E. Berriochoa} and \textit{A. Cachafeiro}, Rend. Mat. Appl., VII. Ser. 19, No. 1, 89-106 (1999; Zbl 0947.42018)].
Mathematics Subject Classification ID: 42C05
Mathematics Subject Classification ID: 33C45
zbMATH DE Number: 1334859
zbMATH Keywords: Sobolev orthogonal polynomials
zbMATH Keywords: asymptotics
zbMATH Keywords: three-term recurrence relation
zbMATH Keywords: norms
zbMATH Keywords: rational modification
author: E. Berriochoa
author: A. Cachafeiro
reviewed by: Andrei Martínez Finkelshtein
MaRDI profile type: Publication
cites work: Orthogonal polynomials of Sobolev type on the unit circle
cites work: Trigonometric Approximation in the Sobolev Spaces $W^{r,2} [ - \pi ,\pi ]$ with Constant Weights
cites work: On polynomials orthogonal with respect to Sobolev inner product on the unit circle
cites work: Orthogonal polynomials on the unit circle and their derivatives
cites work: Q4861743
cites work: Q4061445
full work available at URL: s0377-0427(99)00040-0
OpenAlex ID: W1982995343
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description: scientific article; zbMATH DE number 1334859