Hadamard Product and Resurgence Theory

Abstract

We discuss the analytic continuation of the Hadamard product of two holomorphic functions under assumptions pertaining to Écalle's Resurgence Theory, proving that if both factors are endlessly continuable with prescribed sets of singular points A and B, then so is their Hadamard product with respect to the set {0} ∪ A • B. This is a generalization of the classical Hadamard Theorem in which all the branches of the multivalued analytic continuation of the Hadamard product are considered.

On the Moyal Star Product of Resurgent Series

“…f and g have positive radius of convergence R f and R g correspondingly, then f g ∈ C{ξ} and R f g ≤ R f R g . The following theorem is related to the classical "Hadamard multiplication theorem", and is in fact a weaker version of a theorem proved in [LSS20].…”

“…We will adapt these techniques to our more intricate situation in Section 5. The analytic continuation of the Hadamard product of two Ω-continuable germs has been treated in [LSS20], with a possibly infinite singular locus Ω; our situation is simpler inasmuch as it involves only finite singular loci in the Borel plane, as we will see in Section 6 devoted to the Hadamard part of the formula for * S .…”