Global Rational Approximations of Functions With Factorially Divergent Asymptotic Series

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).