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...

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Bibliographic Details
Published in:Journal d'analyse mathématique (Jerusalem) 2017-10, Vol.133 (1), p.27-49
Main Authors: Sauzin, David, Tiozzo, Giulio
Format: Article
Language:English
Subjects:
Abstract Harmonic Analysis
Analysis
Analytic functions
Dynamical Systems and Ergodic Theory
Functional Analysis
Mathematical analysis
Mathematics
Mathematics and Statistics
Partial Differential Equations
Power series
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Summary: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 Poincaré 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.
ISSN:0021-7670
1565-8538
DOI:10.1007/s11854-017-0026-3