A direct proof of Stahl's theorem for a generic class of algebraic functions

TL;DR

This paper provides a direct, maximum principle-based proof of Stahl's theorem for a generic class of algebraic functions, confirming the zeros distribution of Padé polynomials and convergence of Padé approximants without using orthogonality.

Contribution

It offers a new, straightforward proof of Stahl's theorem for algebraic functions with first-order branch points, avoiding orthogonality relations.

Findings

Confirmed zeros distribution of Padé polynomials

Established convergence in capacity of Padé approximants

Provided a proof applicable to a generic class of algebraic functions

Abstract

Under the assumption of the existence of Stahl's $S$-compact set we give a short proof of the limit zeros distribution of Pad\'e polynomials and convergence in capacity of diagonal Pad\'e approximants for a generic class of algebraic functions. The proof is direct but not from the opposite as Stahl's original proof is. The generic class means in particular that all branch points of the multi-sheeted Riemann surface of the algebraic function are of the first order (i.e., we assume the surface is such that all branch points are of square root type). We do not use the relations of orthogonality at all. The proof is based on the maximum principle only.