  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).
