52 research outputs found
Generalised continuation by means of right limits
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.Comment: 26 page
- …