Generic Approximation of functions by their pad\'{e} approximants, II

TL;DR

This paper extends previous work on approximating functions with Padé approximants by focusing on smooth functions in specific spaces, demonstrating uniform approximation on compact sets.

Contribution

It proves that functions in the space $A^{ abla}( abla)$ can be uniformly approximated by their Padé approximants, including boundary smoothness considerations.

Findings

Approximation by Padé approximants is possible for functions in $A^{ abla}( abla)$.

Uniform convergence on compacta is achieved for smooth boundary functions.

The results extend previous generic approximation theorems to more specific function spaces.

Abstract

In \cite{5} we proved that generically functions defined in any open set can be approximated by a sequense of their pad\'{e} approximants, in the sense of uniform convergence on compacta. In this paper we examine a more particular space, $A^{\infty}(\Omega)$, and prove that we can obtain similar approximation results with functions smooth on the boundary.