Uniformization and Constructive Analytic Continuation of Taylor Series

We analyze the problem of global reconstruction of functions as accurately as possible, based on partial information in the form of a truncated power series at some point and additional analyticity properties. Within the class of functions analytic on a common Riemann surface Omega, with a common rate of growth and a common Maclaurin polynomial, we prove an optimality result on their reconstruction at arbitrary points in Omega, and find a procedure to attain it. This procedure uses the uniformization theorem; the optimal reconstruction errors depend only on the conformal distance to the origin. A priori knowledge of Omega is rigorously available for functions often encountered in analysis (such as solutions of meromomorphic ODEs and classes of PDEs). It is also available, rigorously or conjecturally based on numerical evidence, for perturbative expansions in quantum mechanics, statistical physics and quantum field theory, and in other areas in physics. For a subclass of such functions, we provide the optimal procedure explicitly. These include the Borel transforms of the linear special functions. We construct, in closed form, the uniformization map and optimal procedure for the Borel plane of nonlinear special functions, tronquee solutions of the Painleve equations P_I--P_V. For the latter, $Omega$ is the covering of CZ by curves with fixed origin, modulo homotopies. We obtain some of the uniformization maps as rapidly convergent limits of compositions of elementary maps. Given further information about the function, such as is available for the ubiquitous class of resurgent functions, significantly better approximations are possible and we construct them. In particular, any chosen one of their singularities can be eliminated by specific linear operators which we introduce, and the local structure at the chosen singularity can be obtained in fine detail.