Analytic approximation of matrix functions in Lp

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.

Analytic balayage of measures, Carathéodory domains, and badly approximable functions in Lp

“…1.2]). The problem of describing p-badly approximable functions (in the above context as well as in certain related settings) is considered, for instance, in [1], [3], [4], [8], [13], [14], [17].…”