Orthogonal systems and convolution operators
type: Item
instance of: scholarly article
title: Orthogonal systems and convolution operators
zbMATH Open document ID: 1031.42021
author: Robert L. Ellis
author: I. Gohberg
published in: Operator Theory: Advances and Applications
publication date: 2003-01-28T00:00:00Z
review text: The monograph by R. Ellis and I. Gohberg is devoted to orthogonal polynomials on the unit circle with indefinite inner product and their generalizations. The authors start with the Krein theorem concerning the distribution of zeros of polynomials orthogonal with respect to nonpositive weight function and their continuous analogues. They explore various generalizations and extensions of the theory and consider a more general problem of orthogonalization in modules over \(C^*\)-algebras. One of the main features of the book is the interplay between orthogonal polynomials and generalizations on the one hand, and operator theory on the other hand, especially the theory of Toeplitz matrices, Wiener-Hopf operators, and Fredholm operators and their indices. Various inertia theorems and extended Rouché's theorems come in frequently. In the same way as orthogonal polynomials are connected with Toeplitz matrices, the continuous analogues of orthogonal polynomials are connected with convolution operators on a finite interval. The study of such operators with a focus on invertibility conditions and an inversion formula is one of the key subjects of the book. The book is of interest to both engineers and specialists in analysis.
Mathematics Subject Classification ID: 42C05
Mathematics Subject Classification ID: 47B35
Mathematics Subject Classification ID: 33C47
Mathematics Subject Classification ID: 42-02
Mathematics Subject Classification ID: 47-02
Mathematics Subject Classification ID: 33-02
zbMATH DE Number: 1860219
zbMATH Keywords: orthogonal polynomials
zbMATH Keywords: indefinite inner product
zbMATH Keywords: Krein's theorem
zbMATH Keywords: modules over \(C^*\)-algebras
zbMATH Keywords: Toeplitz matrices and operators
zbMATH Keywords: Wiener-Hopf operators
reviewed by: Leonid Golinskii
MaRDI profile type: Publication
Recommended article: Q3795535
Recommended article: Q5845597
Recommended article: A certain class of orthogonal convergence systems
Recommended article: Q3800514
Recommended article: Orthogonal operators in dynamical stochastic systems
Recommended article: Convolution structures associated with orthogonal polynomials
Recommended article: Q3805252
Recommended article: Q3315650
Recommended article: Q5476808
description: scientific article; zbMATH DE number 1860219