Analytic approximation of matrix functions in $L^p$

TL;DR

This paper develops a comprehensive theory for approximating matrix functions in $L^p$ spaces by analytic functions, introducing new classifications, formulas, and the concept of $p$-superoptimal approximation with uniqueness results.

Contribution

It introduces the classification of matrix functions into respectable and weird, provides explicit formulas for approximation distances, and defines $p$-superoptimal approximation with proven uniqueness.

Findings

Distance to analytic functions equals Hankel operator norm for respectable functions.

Characterization of $p$-badly approximable matrix functions.

Existence and uniqueness of $p$-superoptimal approximants for rational matrix functions.

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 $L^p$, $2\le p<\be$. For an $m\times n$ matrix function $\Phi$ in $L^p$, we consider the Hankel operator $H_\Phi:H^q(C^n)\to H^2_-(C^m)$, $1/p+1/q=1/2$. It turns out that the space of $m\times n$ matrix functions in $L^p$ splits into two subclasses: the set of respectable matrix functions and the set of weird matrix functions. If $\Phi$ is respectable, then its distance to the set of analytic matrix functions is equal to the norm of $H_\Phi$. 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.