Iterated convolutions and endless Riemann surfaces

Abstract

We discuss a version ofÉcalle's definition of resurgence, based on the notion of endless continuability in the Borel plane. We relate this with the notion of Ω-continuability, where Ω is a discrete filtered set or a dicrete doubly filtered set, and show how to construct a universal Riemann surface X Ω whose holomorphic functions are in one-to-one correspondence with Ωcontinuable functions. We then discuss the Ω-continuability of convolution products and give estimates for iterated convolutions of the formφ 1 * · · · * φ n . This allows us to handle nonlinear operations with resurgent series, e.g. substitution into a convergent power series.

Variations on the Resurgence of the Gamma Function

“…Being endlessly continuable, roughly speaking, means that ∧ ϕ(ξ) analytically continues to a (possibly multivalued) holomorphic function without any natural boundary: the analytic continuation is possible along any path except for a discrete set of singularities along the path, that can be circumvented, at will, to the left or to the right (with some stipulations-see [4] for the most general definition, or [2,7]).…”

“…The space of resurgent series is stable under multiplication ( § 2.7) and nonlinear operations like composition or substitution into a convergent series [4], [11], [7]. On the other hand, nonlinear analysis is also compatible with Borel-Laplace summation, but that is much more elementary [11, § 3.5].…”

Hadamard Product and Resurgence Theory

“…The simplest examples of Ω-continuable germs are provided by meromorphic functions and algebraic functions; however, the definition puts no restriction on the nature of the singularities of the analytic continuation of an Ω-continuable germ, only on their location. More general is the definition of "endless continuability" [CNP93], [KS20], and even more general that of "continuability without a cut" [Eca85].…”

“…A benefit of such a detailed rigorous proof with respect to the arguments given in [Eca81] or [CNP93] is that it allows for quantitative estimates. These estimates, in turn, can be used to show the stability of the space of resurgent series under nonlinear operations and not only multiplication [Sau15], [MS16], [KS20].…”

“…Another avenue of research that naturally suggests itself would be to consider the Hadamard product of endlessly continuable germs f and g which are not Ω-continuable in the sense of Definition 1.1 for any discrete closed set Ω. One may conjecture that f ⊙ g is still endlessly continuable because, during the process of analytic continuation by means of contour deformation, one "feels" the presence of only finitely many singularities of f and g at the same time, which is why (additive) convolution (and even iterated convolutions) could be handled in [KS20]; one might try to apply similar techniques to the Hadamard product.…”