Generalised continuation by means of right limits

TL;DR

This paper explores the concept of rrl-continuation for holomorphic functions with natural boundary, proving uniqueness for Poincaré simple pole series and relating it to monogenic regularity.

Contribution

It introduces the notion of rrl-continuation for functions with natural boundaries and proves its uniqueness for Poincaré simple pole series, linking it to monogenic regularity.

Findings

Any Poincaré simple pole series admits a unique rrl-continuation.

The rrl-continuation coincides with the sum of the pole expansion outside the disc.

The work relates rrl-continuation to monogenic regularity in Borel's sense.

Abstract

Several theories have been proposed to generalise the concept of analytic continuation to holomorphic functions of the disc for which the circle is a natural boundary. Elaborating on Breuer-Simon's work on "right limits" of power series, Baladi-Marmi-Sauzin recently introduced the notion of "renascent right limit" and "rrl-continuation". We discuss a few examples and consider particularly the classical example of "Poincar{\'e} simple pole series" in this light. These functions are represented in the disc as series of infinitely many simple poles located on the circle; they appear for instance in small divisor problems in dynamics. We prove that any such function admits a unique rrl-continuation, which coincides with the function obtained outside the disc by summing the simple pole expansion. We also discuss the relation with monogenic regularity in the sense of Borel.