Generalised continuation by means of right limits

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.