Explicit almost-periodic factorization for a class of triangular matrix functions
type: Item
instance of: scholarly article
title: Explicit almost-periodic factorization for a class of triangular matrix functions
zbMATH Open document ID: 1183.47011
author: M. C. Martins
author: M. C. Câmara
published in: Journal d'Analyse Mathématique
publication date: 2008-05-27T00:00:00Z
review text: Let \(APW\) be the Banach algebra of almost periodic series \[ f=\sum_j c_j e^{ix\lambda}\quad\text{with}\;\;\lambda_j\in\mathbb{R},\;c_j\in\mathbb{C}\;\;\text{and}\;\;\|f\|_W=\sum_j |c_j|<\infty, \] and let \(APW^\pm\) be the Banach subalgebras of \(APW\) constituted by functions with \(\pm\lambda_j\geq 0\), respectively. An almost periodic matrix function \(G\) admits a (right) canonical \(APW\) factorization if it is represented in the form \(G=G_-G_+\), where the entries of the matrix functions \(G_\pm, G_\pm^{-1}\) are in \(APW^\pm\), respectively. Under conditions of existence, a canonical \(APW\) factorization, which automatically is a canonical generalized factorization relative to \(L_2(\mathbb{R})\), is constructed explicitly for triangular almost periodic matrix function of the form \[ G=\begin{bmatrix} E^{-1}&0\\ a E^\alpha+b+cE^{-\beta}&E\end{bmatrix}, \] where \(E(\xi)=e^{i\xi}\) for \(\xi\in\mathbb{R}\), \(\alpha,\beta\in(0,1)\), \(\alpha/\beta\notin\mathbb{Q}\), \(a,b,c\in\mathbb{C}\setminus\{0\}\), and either (i) \(n\alpha+\beta=1\) for some \(n\in\mathbb{N}\) and \(\beta>(n-1)\alpha\), or (ii) \(2\alpha+\beta=1\) and \(\beta<\alpha\). In case (i), the matrix function \(G\) admits a canonical \(APW\) factorization if and only if \(|c/b||a/b|^{\beta/\alpha}\neq 1/n\). In case (ii), such a factorization is constructed under the following sufficient conditions: either \(|c/b||a/b|^{\beta/\alpha}<1/2\), or \(|c/b||a/b|^{\beta/\alpha}>m/(m+1)\), where \(m=\big[\frac{\alpha}{\beta}\big]+1\). The existence of almost periodic polynomial solutions is also studied.
reviewed by: Yuri I. Karlovich
Mathematics Subject Classification ID: 47A68
Mathematics Subject Classification ID: 42A75
Mathematics Subject Classification ID: 30E25
zbMATH DE Number: 5279121
zbMATH Keywords: canonical almost periodic factorization
zbMATH Keywords: almost periodic matrix function
zbMATH Keywords: Riemann-Hilbert problem
MaRDI profile type: Publication
full work available at URL: s11854-008-0007-7
OpenAlex ID: W1989763830
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DOI: 10.1007/S11854-008-0007-7
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description: scientific article; zbMATH DE number 5279121