Multipoint Pad\'e Approximants to Complex Cauchy Transforms with Polar Singularities

TL;DR

This paper investigates the convergence properties of multipoint Padé approximants to complex Cauchy transforms with polar singularities, revealing how poles and approximants behave under specific conditions.

Contribution

It provides new results on the convergence of pole distributions and approximants for complex measures with rational perturbations, extending classical approximation theory.

Findings

Poles of approximants converge to the balayage of the conjugate-symmetric distribution.

Approximants converge in capacity outside the measure's support.

Poles of the rational function attract at least as many poles of the approximants as their multiplicity.

Abstract

We study diagonal multipoint Pad\'e approximants to sums of a Cauchy transform of a complex measure and a rational function. The measure is assumed to have compact regular support included into the real line and an argument of bounded variation on the support. For interpolation sets whose normalized counting measures converge sufficiently fast in the weak-star sense to some conjugate-symmetric distribution, we show that the counting measures of poles of the approximants converge to the balayage of that distribution onto the support of the measure, in the weak-star sense, that the approximants themselves converge in capacity to the approximated function outside the support of the measure, and that the poles of the additional rational function attract at least as many poles of the approximants as their multiplicity and not much more.