Analytic approximation of matrix functions and dual extremal functions

TL;DR

This paper investigates conditions for the existence of dual extremal functions in matrix approximation problems, linking them to Hankel operators and special factorizations involving thematic matrix functions.

Contribution

It provides a characterization of matrix functions that admit dual extremal functions using Hankel operators and thematic matrix factorizations.

Findings

Existence of dual extremal functions is characterized by maximizing vectors of Hankel operators.

A connection is established between dual extremal functions and specific matrix factorizations.

The results offer a new perspective on matrix function approximation on the unit circle.

Abstract

We study the question of the existence of a dual extremal function for a bounded matrix function on the unit circle in connection with the problem of approximation by analytic matrix functions. We characterize the class of matrix functions, for which a dual extremal function exists in terms of the existence of a maximizing vector of the corresponding Hankel operator and in terms of certain special factorizations that involve thematic matrix functions.