Minimax principle and lower bounds in H$^{2}$-rational approximation

TL;DR

This paper establishes lower bounds for rational approximation in the Hardy space H^2, applying them to Blaschke products and Cauchy integrals, and discusses computational methods with numerical experiments.

Contribution

It introduces new lower bounds for H^2 rational approximation and explores their computation via Adamjan-Arov-Krein theory and linearized errors.

Findings

Derived lower bounds for approximation errors in H^2

Applied bounds to Blaschke products and Cauchy integrals

Presented numerical experiments validating the bounds

Abstract

We derive some lower bounds in rational approximation of given degree to functions in the Hardy space $H^2$ of the disk. We apply these to asymptotic errors rates in approximation to Blaschke products and to Cauchy integrals on geodesic arcs. We also explain how to compute such bounds, either using Adamjan-Arov-Krein theory or linearized errors, and we present a couple of numerical experiments on several types of functions. We dwell on the Adamjan-Arov-Krein theory and a maximin principle developed in the article "An L^p analog of AAK theory for p \textgreater{}= 2", by L. Baratchart and F. Seyfert, in the Journal of Functional Analysis, 191 (1), pp. 52-122, 2012.