Meromorphic Approximants to Complex Cauchy Transforms with Polar Singularities

TL;DR

This paper investigates meromorphic approximants to functions combining rational parts and Cauchy transforms with polar singularities, revealing convergence properties and pole distribution behaviors in complex approximation theory.

Contribution

It provides new results on the convergence and pole distribution of meromorphic approximants to complex Cauchy transforms with polar singularities.

Findings

Pole counting measures converge to Green equilibrium distribution.

Approximants converge in capacity to the target function.

Poles of the rational part attract at least as many poles of the approximants.

Abstract

We study AAK-type meromorphic approximants to functions $F$, where $F$ is a sum of a rational function $R$ and a Cauchy transform of a complex measure $\lambda$ with compact regular support included in $(-1,1)$, whose argument has bounded variation on the support. The approximation is understood in $L^p$-norm of the unit circle, $p\geq2$. We obtain that the counting measures of poles of the approximants converge to the Green equilibrium distribution on the support of $\lambda$ relative to the unit disk, that the approximants themselves converge in capacity to $F$, and that the poles of $R$ attract at least as many poles of the approximants as their multiplicity and not much more.