Nuttall's theorem with analytic weights on algebraic S-contours

TL;DR

This paper extends Nuttall's theorem to provide asymptotic error formulas for Padé approximants of Cauchy integrals with analytic densities on algebraic S-contours, broadening its applicability.

Contribution

It generalizes Nuttall's theorem to algebraic S-contours, allowing for asymptotic analysis of Padé approximants in more complex contour settings.

Findings

Extended Nuttall's theorem to algebraic S-contours.

Derived asymptotic error formulas for Padé approximants.

Applicable to Cauchy integrals of analytic densities.

Abstract

Given a function $f$ holomorphic at infinity, the $n$-th diagonal Pad\'e approximant to $f$, denoted by $[n/n]_f$, is a rational function of type $(n,n)$ that has the highest order of contact with $f$ at infinity. Nuttall's theorem provides an asymptotic formula for the error of approximation $f-[n/n]_f$ in the case where $f$ is the Cauchy integral of a smooth density with respect to the arcsine distribution on [-1,1]. In this note, Nuttall's theorem is extended to Cauchy integrals of analytic densities on the so-called algebraic S-contours (in the sense of Nuttall and Stahl).