On the continuous analog of Rakhmanov's theorem for orthogonal polynomials.
type: Item
instance of: scholarly article
title: On the continuous analog of Rakhmanov's theorem for orthogonal polynomials.
zbMATH Open document ID: 1041.34075
DOI: 10.1016/S0022-1236(02)00073-3
author: Sergey Denisov
published in: Journal of Functional Analysis
publication date: 2003-04-28T00:00:00Z
review text: Consider the Sturm-Liouville operator in \(L^2(0, \infty)\) defined by \[ H_\alpha f=-f'' +q f, \quad f(0)=\alpha f'(0),\; \alpha\in {\mathbb R}\cup \{\infty\}, \] where \(q\) is a uniformly locally bounded real-valued function. The main result of the paper says that, if the essential spectrum and the absolutely continuous component of the spectral measure fill \((0, \infty)\), then it holds \[ \lim_{x\to \infty}\int_{x-\delta}^{x+\delta}q(t)\,dt\to 0 \] for all \(\delta>0\). To prove this result, the author defines a continuous analog of orthogonal polynomials on the unit circle (associated with a measure), and, for this new polynomials, he establishes a variant of Rakhmanov' s theorem.
Mathematics Subject Classification ID: 34L05
Mathematics Subject Classification ID: 42C05
Mathematics Subject Classification ID: 47E05
zbMATH DE Number: 1901777
zbMATH Keywords: Sturm-Liouville operator
zbMATH Keywords: orthogonal polynomials
zbMATH Keywords: Rakhmanov' s theorem
zbMATH Keywords: Krein system
reviewed by: Ingrid Beltiţă
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1901777