Inverse Theorem on Row Sequences of Linear Pad\'e-orthogonal Approximation

TL;DR

This paper establishes necessary and sufficient conditions for the geometric convergence of denominators in linear Padé-orthogonal approximants for measures on complex compact sets, extending Gonchar's theorem.

Contribution

It provides an analogue of Gonchar's theorem for row sequences of Padé-orthogonal approximants, detailing convergence criteria in a general setting.

Findings

Conditions for geometric convergence of denominators

Extension of Gonchar's theorem to orthogonal approximants

Applicable to measures on arbitrary compact sets in the complex plane

Abstract

We give necessary and sufficient conditions for the convergence with geometric rate of the denominators of linear Pad\'e-orthogonal approximants corresponding to a measure supported on a general compact set in the complex plane. Thereby, we obtain an analogue of Gonchar's theorem on row sequences of Pad\'e approximants.