Analytic approximation of matrix functions in [formula omitted

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 < ∞ . For an m × n matrix function Φ in L p , we consider the Hankel operator H Φ : H q ( C n ) → H − 2 ( 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.