Global Rational Approximations of Functions With Factorially Divergent Asymptotic Series

TL;DR

This paper introduces dyadic expansions as a new convergent and asymptotic representation method for functions with factorially divergent series, providing efficient, uniform, and geometrically convergent expansions applicable to special functions and operators.

Contribution

The paper develops dyadic expansions that are convergent and asymptotic, extending their applicability to special functions and operators, including resurgent functions and resolvents.

Findings

Dyadic expansions are geometrically convergent and numerically efficient.

They extend to complex plane regions minus a chosen ray.

Applicable to special functions and operator resolvents.

Abstract

We construct a new type of convergent, and asymptotic, representations, dyadic expansions. Their convergence is geometric and the region of convergence often extends from infinity down to $0^+$. We show that dyadic expansions are numerically efficient representations. For special functions such as Bessel, Airy, Ei, erfc, Gamma, etc. the region of convergence of dyadic series is the complex plane minus a ray, with this cut chosen at will. Dyadic expansions thus provide uniform, geometrically convergent asymptotic expansions including near antistokes rays. We prove that relatively general functions, \'Ecalle resurgent ones, possess convergent dyadic expansions. These expansions extend to operators, resulting in representations of the resolvent of self-adjoint operators as series in terms of the associated unitary evolution operator evaluated at some prescribed discrete times (alternatively, for positive operators, in terms of the generated semigroup).