On Uniform Approximation of Rational Perturbations of Cauchy Integrals

TL;DR

This paper investigates the convergence properties of AAK and Padé approximants for functions combining Cauchy transforms of measures and rational functions, under specific regularity conditions.

Contribution

It establishes convergence results for these approximants when approximating functions with particular measure and rational function properties.

Findings

AAK and Padé approximants converge locally uniformly in the domain of holomorphy.

Convergence holds in the unit disk for AAK approximants.

Padé approximants require a nearly conjugate-symmetric interpolation scheme.

Abstract

We study AAK as well as Pad\'e approximants 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. It is shown that the approximants converge to f locally uniformly in the domain of holomorphy of f, intersected with the unit disk in the case of AAK approximants. In the case of Pad\'e approximants we need to assume that the interpolation scheme is "nearly" conjugate-symmetric.