Best rational approximation of functions with logarithmic singularities

TL;DR

This paper investigates the asymptotic behavior of best rational approximations to functions with logarithmic singularities on the unit circle, using advanced operator theory techniques.

Contribution

It provides an asymptotic formula for the approximation error in the BMO-norm, extending understanding of rational approximation for functions with singularities.

Findings

Derived an asymptotic formula for the approximation error

Connected approximation quality with singular values of Hankel operators

Extended classical approximation results to functions with logarithmic singularities

Abstract

We consider functions $\omega$ on the unit circle $\mathbb T$ with a finite number of logarithmic singularities. We study the approximation of $\omega$ by rational functions and find an asymptotic formula for the distance in the BMO-norm between $\omega$ and the set of rational functions of degree $n$ as $n\to\infty$. Our approach relies on the Adamyan-Arov-Krein theorem and on the study of the asymptotic behaviour of singular values of Hankel operators.