Universal Pad\'e approximants on simply connected domains

TL;DR

This paper extends the concept of universal approximation from Taylor series to Padé approximants, demonstrating their ability to approximate holomorphic functions on certain compact sets outside their domain.

Contribution

It provides the first generic result showing universal approximation by Padé approximants for functions in simply connected domains, broadening the scope beyond polynomial approximations.

Findings

Padé approximants can universally approximate functions outside their domain

Universal functions can be smooth on the boundary if sets are disjoint from the domain boundary

The results apply to simply connected domains with connected complement sets

Abstract

The theory of universal Taylor series can be extended to the case of Pad\'e approximants where the universal approximation is not realized by polynomials any more, but by rational functions, namely the Pad\'e approximants of some power series. We present the first generic result in this direction, for Pad\'e approximants corresponding to Taylor developments of holomorphic functions in simply connected domains. The universal approximation is required only on compact sets $K$ which lie outside the domain of definition and have connected complement. If the sets $K$ are additionally disjoint from the boundary of the domain of definition, then the universal functions can be smooth on the boundary.