Analytic approximation of matrix functions in L^p

2008/05/28 by L. Baratchart, Laurent Baratchart, Baratchart, L. +6
Mathematics · #30D55 #30E10 #47B35 #Approximation Theory and Sequence Spaces #Classical Analysis and ODEs (math.CA) #Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Mathematical functions and polynomials #math.CA #math.CV #math.FA #msc:30D55 #msc:30E10 #msc:47B35

paper · pdf · doi:10.48550/arxiv.0805.4366

43 pages

arxiv created 2008/05/28 · openalex publication_date 2008/05/28 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the problem of approximation of matrix functions of class L^p on the unit circle by matrix functions analytic in the unit disk in the norm of L^p, 2≤ p<\be. For an m× n matrix function Φ in L^p, we consider the Hankel operator H_Φ:H^q(Cⁿ)→ H²_-(C^m), 1/p+1/q=1/2. It turns out that the space of m× n matrix functions in L^p splits into two subclasses: the set of respectable matrix functions and the set of weird matrix functions. If Φ is respectable, then its distance to the set of analytic matrix functions is equal to the norm of H_Φ. For weird matrix functions, to obtain the distance formula, we consider Hankel operators defined on spaces of matrix functions. We also describe the set of p-badly approximable matrix functions in terms of special factorizations and give a parametrization formula for all best analytic approximants in the norm of L^p. Finally, we introduce the notion of p-superoptimal approximation and prove the uniqueness of a p-superoptimal approximant for rational matrix functions.

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