On factorization of trigonometric polynomials
type: Item
instance of: scholarly article
title: On factorization of trigonometric polynomials
zbMATH Open document ID: 1070.47011
author: Michael A. Dritschel
published in: Integral Equations and Operator Theory
publication date: 2004-10-28T00:00:00Z
review text: In the present paper, using only ideas from elementary operator theory, a new proof of the operator version of the Fejer-Riesz theorem is given, and some of the ramifications of the ideas of the proof are studied. Starting with a sketch of some basic results on Schur complements and factorization, some simpler proofs are given. Moreover, Schur complements are applied to the study of doubly infinite Toeplitz operators with operator coefficients. A proof of the operator Fejer-Riesz theorem is given in the second section of the paper, and in the third one an explicit construction of the outer factorization is given. Also, several interesting problems are raised by this construction. In the fourth section, a characterization of outer factorizations and extremals is presented, and in the fifth one it is shown how the proof given of the operator Fejer-Riesz theorem can be extended to study the factorization of trigonometric polynomials in several variables. Since even in this case, there are numerous examples of positive polynomials which cannot be factored in terms of polynomials, in the sixth section some sufficient conditions for factorization of multivariate trigonometric polynomials are given. In the last section of the paper, a number of families of operators are constructed which lie somewhere between the contractions and numerical radius contractions, and which are also described via positive trigonometric polynomials.
reviewed by: Ilie Valuşescu
Mathematics Subject Classification ID: 47A68
Mathematics Subject Classification ID: 47A65
Mathematics Subject Classification ID: 42A05
Mathematics Subject Classification ID: 30C10
Mathematics Subject Classification ID: 15A23
zbMATH DE Number: 2111520
zbMATH Keywords: operator theory
zbMATH Keywords: trigonometric polynomials
zbMATH Keywords: factorization
zbMATH Keywords: multivariate polynomials
zbMATH Keywords: linear matrix inequalities
zbMATH Keywords: Fejer-Riesz theorem
zbMATH Keywords: Schur complement
zbMATH Keywords: Cesàro means
zbMATH Keywords: Agler families
Wikidata QID: Q57451941
MaRDI profile type: Publication
full work available at URL: s00020-002-1198-4
OpenAlex ID: W2030224778
DOI: 10.1007/S00020-002-1198-4
description: scientific article; zbMATH DE number 2111520