On $L_{\mathbb R}^2$-best rational approximants to Markov functions on several intervals

TL;DR

This paper investigates the asymptotic behavior of the error in the best rational approximation of Markov functions supported on multiple intervals, focusing on $L^2$-norm on the unit circle.

Contribution

It provides new insights into the strong asymptotics of $L^2$-best rational approximants for Markov functions supported on several intervals.

Findings

Asymptotic error behavior characterized for large $n$

Extension of classical approximation results to multiple intervals

Detailed analysis of poles inside the unit disk

Abstract

Let $ f(z)=\int(z-x)^{-1}d\mu(x) $, where $ \mu $ is a Borel measure supported on several subintervals of $ (-1,1) $ with smooth Radon-Nikodym derivative. We study strong asymptotic behavior of the error of approximation $ (f-r_n)(z) $, where $ r_n(z) $ is the $ L_{\mathbb R}^2$-best rational approximant to $ f(z) $ on the unit circle with $ n $ poles inside the unit disk.