Asymptotic Uniqueness of Best Rational Approximants to Complex Cauchy Transforms in ${L}^2$ of the Circle

TL;DR

This paper proves that for sufficiently large degrees, the best rational approximants to certain complex Cauchy transforms are uniquely determined in the L^2 sense on the unit circle, under specific regularity conditions.

Contribution

It establishes the asymptotic uniqueness of critical points in best rational approximation for a class of functions involving Cauchy transforms, extending previous results to a broader setting.

Findings

Uniqueness of critical points for large n in rational approximation

Conditions on the measure's Radon-Nikodym derivative ensure uniqueness

Results apply to functions with Cauchy transforms supported on real intervals

Abstract

For all n large enough, we show uniqueness of a critical point in best rational approximation of degree n, in the L^2-sense on the unit circle, to functions f, where f is a sum of a Cauchy transform of a complex measure \mu supported on a real interval included in (-1,1), whose Radon-Nikodym derivative with respect to the arcsine distribution on its support is Dini-continuous, non-vanishing and with and argument of bounded variation, and of a rational function with no poles on the support of \mu.