Pad\'e approximants to certain elliptic-type functions

TL;DR

This paper derives strong asymptotics for diagonal Padé approximants to a class of elliptic-type functions defined via Cauchy integrals over a minimal capacity continuum, analyzing convergence and spurious poles.

Contribution

It provides new asymptotic formulas for Padé approximants to elliptic-type functions and describes their convergence behavior and spurious pole distribution.

Findings

Asymptotic behavior of Padé approximants is characterized.

Regions of uniform convergence are identified.

Spurious poles are analyzed in a generic setting.

Abstract

Given non-collinear points a_1, a_2, a_3, there is a unique compact, say \Delta, that has minimal logarithmic capacity among all continua joining a_1, a_2, and a_3. For h be a complex-valued non-vanishing Dini-continuous function on \Delta, we consider f_h(z) := (1/\pi i)\int_\Delta h(t)/(t-z) dt/w^+(t), where w(z) := \sqrt{\prod_{k=0}^3(z-a_k)} and w^+ the one-sided value according to some orientation of \Delta. In this work we present strong asymptotics of diagonal Pad\'e approximants to f_h and describe the behavior of the spurious pole and the regions of locally uniform convergence from a generic perspective.