Iterated convolutions and endless Riemann surfaces

TL;DR

This paper explores the concept of endless continuability in the Borel plane, constructing a universal Riemann surface to analyze resurgent functions and their convolutions, enabling better handling of nonlinear operations in resurgence theory.

Contribution

It introduces a universal Riemann surface for alle's resurgence framework and provides estimates for iterated convolutions, advancing the analysis of nonlinear operations on resurgent series.

Findings

Constructed a universal Riemann surface X_alle for alle-continuable functions.

Established estimates for iterated convolutions of resurgent functions.

Linked alle's resurgence with alle-continuability and convolution operations.

Abstract

We discuss a version of \'Ecalle's definition of resurgence, based on the notion of endless continuability in the Borel plane. We relate this with the notion of \Omega-continuability, where \Omega\ is a discrete filtered set, and show how to construct a universal Riemann surface X_\Omega\ whose holomorphic functions are in one-to-one correspondence with \Omega-continuable functions. We then discuss the \Omega-continuability of convolution products and give estimates for iterated convolutions of the form \hat\phi_1*\cdots *\hat\phi_n. This allows us to handle nonlinear operations with resurgent series, e.g. substitution into a convergent power series.