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arXiv · 1610.05453

Iterated convolutions and endless Riemann surfaces

Shingo Kamimoto·David Sauzin
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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.

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Shingo Kamimoto, David Sauzin. 2016-10-18. Iterated convolutions and endless Riemann surfaces. https://arxiv.org/abs/1610.05453

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