Hamburger moment problem: Parseval equality and \(A\)-singularity
type: Item
instance of: scholarly article
title: Hamburger moment problem: Parseval equality and \(A\)-singularity
zbMATH Open document ID: 0917.47015
author: Alexander Kheifets
published in: Journal of Functional Analysis
publication date: 1999-07-27T00:00:00Z
review text: To solve the Nehari problem, one has to find all \(L^\infty\) functions on the unit circle, bounded in modulus by 1, whose Fourier coefficients with negative index are prescribed. All the solutions are of the form \(w=(a/\overline{a})(\omega-\overline{b})/(1-\omega b)\) with \(\omega\) an arbitrary Schur function and \((a,b)\) a pair of functions defined by the data. The pair \((a,b)\) is called a Nehari pair. A Nehari pair has the properties: (1) \(b\) is a Schur function and \(b(0)=0\), (2) \(\log(1-| b| ^2)\in L^1\), \(| a| ^2=1-| b| ^2\), and \(a\) is an outer function of the Schur class, (3) \(b\) has the (ac) property, that means that the Riesz-Herglotz measure associated with the Schur function \(b\omega\) is absolutely continuous for all Schur functions \(\omega\). In this paper it is proved that properties (1)--(3) are necessary but not sufficient for \((a,b)\) to be a Nehari pair. More precisely, it is proved that if \((a,b)\) parametrizes the solutions of an infinite indeterminate Hamburger moment problem, then it satisfies (1)-(3), but it is not a Nehari pair because it is Arov singular. To prove this, the machinary of the abstract interpolation problem is used, which lifts many classical complex interpolation and moment problems to an abstract Hilbert space setting. The Parseval identity from the title refers to a general Parseval identity in this setting which is used to prove the (ac) property of the pair \((a,b)\) when related to the Hamburger moment problem.
reviewed by: Adhemar Bultheel
Mathematics Subject Classification ID: 47A57
Mathematics Subject Classification ID: 30E05
zbMATH DE Number: 950735
zbMATH Keywords: Hamburger moment problem
zbMATH Keywords: abstract interpolation problem
zbMATH Keywords: Parseval equality
zbMATH Keywords: Schur function
zbMATH Keywords: Nehari problem
zbMATH Keywords: Arov singular
zbMATH Keywords: Nehari pair
zbMATH Keywords: Riesz-Herglotz measure
MaRDI profile type: Publication
full work available at URL: jfan.1996.0134
OpenAlex ID: W2003778907
DOI: 10.1006/JFAN.1996.0134
description: scientific article; zbMATH DE number 950735