On Convergence of Rational Hermite-Pad\'e Approximants

TL;DR

This paper compares convergence rates of rational Hermite-Padé and Padé approximants for multivalued analytic functions, showing Hermite-Padé approximants converge faster using scalar potential methods.

Contribution

It introduces a new proof technique based on scalar mixed Green-logarithmic potential problems for convergence analysis.

Findings

Hermite-Padé approximants converge faster than Padé approximants.

The proof uses scalar mixed Green-logarithmic potential problems.

Results apply to a class of multivalued analytic functions.

Abstract

The main purpose of this paper is to compare the convergence properties of Pad\'e approximants and rational Hermite-Pad\'e approximants for some model class of multivalued analytic functions based of the inverse Zhoukovsky transform. We prove that in the class of analytic functions under consideration the rational Hermite-Pad\'e approximants converge faster than the corresponding Pad\'e approximants. In contrast to the classical vector potential-theoretic approach, which was introduced by A. A. Gonchar and E. A. Rakhmanov in 1981 and developed later by A. I. Aptekarev, V. N. Sorokin and others, the proofs here are based on some scalar mixed Green-logarithmic potential problems.