Minimax principle and lower bounds in H$^{2}$-rational approximation
Abstract
We derive some lower bounds in rational approximation of given degree to functions in the Hardy space 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.
Cite
@article{arxiv.1501.01161,
title = {Minimax principle and lower bounds in H$^{2}$-rational approximation},
author = {Laurent Baratchart and Sylvain Chevillard and Tao Qian},
journal= {arXiv preprint arXiv:1501.01161},
year = {2015}
}
Comments
Accepted for publication in the special issue of Journal of Approximation Theory / Matematicheskii Sbornik, to the memory of A. A. Gonchar and H. Stahl