On polynomials orthogonal with respect to Sobolev inner product on the unit circle
type: Item
instance of: scholarly article
title: On polynomials orthogonal with respect to Sobolev inner product on the unit circle
zbMATH Open document ID: 0897.42013
author: Xin Li
author: Francisco Marcellán
published in: Pacific Journal of Mathematics
publication date: 1998-09-28T00:00:00Z
full work available at URL: nyjm.albany.edu
review text: In the present paper the authors investigate orthogonal polynomials \(\psi_n\) with respect to an inner product involving a positive definite measure \(\mu\) on the unit circle to which a Sobolev part is added which involves the first order derivatives of the functions at \(m\) points \(z_1,z_2,\ldots,z_m\) in the complex plane. The Sobolev part is of the form \(( f'(z_1)\cdots f'(z_m)) A( g'(z_1) \cdots g'(z_m))^H\), where \(A\) is a complex \(m\times m\) matrix. It is shown that the corresponding orthogonal polynomials exist for large \(n\) whenever \(A\) is non-singular. Algebraic properties expressing these polynomials \(\psi_n\) and their derivative \(\psi_n'\) in terms of the orthogonal polynomials \(\varphi_n\) for the measure \(\mu\) are given. From this the relative asymptotic behavior of the ratio \(\psi_n(z)/\varphi_n(z)\) is deduced. The methods and results are similar to those used to study Sobolev orthogonal polynomials on the real line.
Mathematics Subject Classification ID: 42C05
zbMATH DE Number: 1025782
zbMATH Keywords: orthogonal polynomials on the unit circle
zbMATH Keywords: Sobolev inner product
reviewed by: Walter Van Assche
MaRDI profile type: Publication
OpenAlex ID: W1601562475
DOI: 10.2140/PJM.1996.175.127
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description: scientific article; zbMATH DE number 1025782