Favard theorem for reproducing kernels
type: Item
instance of: scholarly article
title: Favard theorem for reproducing kernels
zbMATH Open document ID: 0851.30001
DOI: 10.1016/0377-0427(93)E0234-D
author: Adhemar Bultheel
author: Olav Njåstad
author: Pablo González-Vera
author: Erik Hendriksen
published in: Journal of Computational and Applied Mathematics
publication date: 1995-07-04T00:00:00Z
review text: Consider for \(n = 0, 1, \dots\) the nested spaces \({\mathcal L}_n\) of rational functions of degree \(n\) at most with given poles \(1/ \overline \alpha_i\), \(|\alpha_i |< 1\), \(i = 1, \dots, n\). Let \({\mathcal L} = \cup^\infty_0 {\mathcal L}_n\). Given a finite positive measure \(\mu\) on the unit circle, we associate with it an inner product on \({\mathcal L}\) by \(\langle f,g \rangle = \int f \overline gd \mu\). Suppose \(k_n (z,w)\) is the reproducing kernel for \({\mathcal L}_n\) i.e., \(\langle f(z), k_n (z,w) \rangle = f(w)\), for all \(f \in {\mathcal L}_n\), \(|w |< 1\), then it is known that they satisfy a coupled recurrence relation. In this paper we shall prove a Favard type theorem which says that if you have a sequence of kernel functions \(k_n (z,w)\) which are generated by such a recurrence, then there will be a measure \(\mu\) supported on the unit circle so that \(k_n\) is the reproducing kernel for \({\mathcal L}_n\). The measure is unique under certain extra conditions on the points \(\alpha_i\).
reviewed by: Adhemar Bultheel
Mathematics Subject Classification ID: 30C15
zbMATH DE Number: 770335
zbMATH Keywords: orthogonal rational functions
zbMATH Keywords: Favard theorem
zbMATH Keywords: reproducing kernel
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 770335