Connections between the Carathéodory-Toeplitz and the Nehari extension problems: The discrete scalar case
type: Item
instance of: scholarly article
title: Connections between the Carathéodory-Toeplitz and the Nehari extension problems: The discrete scalar case
zbMATH Open document ID: 0971.47009
DOI: 10.1007/BF01192420
author: Daniel Alpay
author: I. Gohberg
published in: Integral Equations and Operator Theory
publication date: 2001-10-18T00:00:00Z
review text: Let \({\mathcal W}\) denote the Wiener algebra of functions defined on the unit circle with Fourier coefficients in \(\ell_1 (\mathbb{Z})\) (i.e., \(f(z) = \sum_{n \in \mathbb{Z}} f_nz^n\), with \(\sum_{n \in \mathbb{Z}} |f_n |< \infty\)), and let us denote by \({\mathcal W}_{+}\) the subalgebra of functions with \(f_n = 0\) for all \(n < 0\) and by \({\mathcal W}_{-}\) the subalgebra of functions with \(f_{n}=0\) for all \(n > 0\). A function \(f \in {\mathcal W}\) is said to have a Weiner-Hopf factorization if it can be written as \(f = f_{+} f_{-}\), where both \(f_{+}\) and \(1/ f_{+}\) belong to \({\mathcal W}_{+}\) and both \(f_{-}\) and \(1/ f_{-}\) belong to \({\mathcal W}_{-}\). The Carathéodory-Toeplitz and the Nehary extension problems (resp.) both consists in the description of the class of functions \(f\in {\mathcal W}\) whose Fourier coefficients satisfy certain restrictions (they interpolate a fixed special sequence of values) and their solution depends on certain sequences of matrix functions \(T_n(z)\) and \(H_n(z)\) (resp.). In the paper a relation is stated which exists between the \(T_n\) and the \(H_n\), which is given by two complicated formulas (see Theorem 3.1 of the paper). The problem is firstly solved for the rational functions which satisfy the corresponding extension problems and then the authors use an approximation theoretical argument to prove the main result in the general case. The paper is continuation of a previous one by the same authors [see: Integral Equations Oper. Theory 26, No. 3, 249-272 (1996; Zbl 0871.47015)].
Mathematics Subject Classification ID: 47A57
Mathematics Subject Classification ID: 47A68
Mathematics Subject Classification ID: 34L25
Mathematics Subject Classification ID: 81U40
Mathematics Subject Classification ID: 42B05
Mathematics Subject Classification ID: 42A85
zbMATH DE Number: 1507904
zbMATH Keywords: Wiener algebra
zbMATH Keywords: extension problems
zbMATH Keywords: Wiener-Hopf factorization
zbMATH Keywords: Carathéodory-Toeplitz problems
zbMATH Keywords: Nehary extension problems
reviewed by: Jose Maria Almira
MaRDI profile type: Publication
cites work: Q4855881
cites work: Inverse spectral problems for difference operators with rational scattering matrix function
cites work: A relationship between the Nehari and the Carathéodory-Toeplitz extension problems
cites work: Infinite analogues of block Toeplitz matrices and related orthogonal functions
cites work: Q3270024
cites work: Classes of linear operators. Vol. II
cites work: Szegö-Kac-Achiezer formulas in terms of realizations of the symbol
full work available at URL: https://doi.org/10.1007/bf01192420
OpenAlex ID: W2026740955
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description: scientific article; zbMATH DE number 1507904