Hermite-Pad\'e approximants for a pair of Cauchy transforms with overlapping symmetric supports

TL;DR

This paper studies the convergence and asymptotic behavior of Hermite-Padé approximants for pairs of Cauchy transforms with overlapping symmetric supports, extending known results to new overlapping support configurations.

Contribution

It derives Szegő-type asymptotic formulas, characterizes convergence/divergence domains, and identifies overinterpolation phenomena for overlapping support cases.

Findings

Szegő-type asymptotics for overlapping supports

Identification of divergence domains absent in Nikishin systems

Presence of overinterpolation in Nikishin systems

Abstract

Hermite-Pad\'e approximants of type II are vectors of rational functions with common denominator that interpolate a given vector of power series at infinity with maximal order. We are interested in the situation when the approximated vector is given by a pair of Cauchy transforms of smooth complex measures supported on the real line. The convergence properties of the approximants are rather well understood when the supports consist of two disjoint intervals (Angelesco systems) or two intervals that coincide under the condition that the ratio of the measures is a restriction of the Cauchy transform of a third measure (Nikishin systems). In this work we consider the case where the supports form two overlapping intervals (in a symmetric way) and the ratio of the measures extends to a holomorphic function in a region that depends on the size of the overlap. We derive Szeg\H{o}-type formulae for the asymptotics of the approximants, identify the convergence and divergence domains (the divergence domains appear for Angelesco systems but are not present for Nikishin systems), and show the presence of overinterpolation (a feature peculiar for Nikishin systems but not for Angelesco systems). Our analysis is based on a Riemann-Hilbert problem for multiple orthogonal polynomials (the common denominator).