Uniformization and Constructive Analytic Continuation of Taylor Series

Ovidiu Costin and Gerald V. Dunne

Department of Mathematics, The Ohio State University, Columbus, OH 43210-1174, USA Department of Physics, University of Connecticut, Storrs, CT 06269-3046, USA

Abstract 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 addi- tional analyticity properties. Within the class of functions analytic on a common Riemann surface Ω, with a common rate of growth and a common Maclaurin polynomial, we prove an optimality result on their reconstruction at arbitrary points in Ω, 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 Ω is rigorously available for functions often encountered in analysis (such as solutions of meromomorphic ODEs and classes of PDEs). It is also available, rigor- ously 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 in- clude 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, tronquée solutions of the Painlevé equations PI–PV. For the latter, Ω is the covering of C \ Z 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 opera- tors which we introduce, and the local structure at the chosen singularity can be obtained in fine detail. MSC codes: 30F20, 30B30, 30B10, 40A25, 41A58 arXiv:2009.01962v3 [math.CV] 14 Apr 2021

1 Introduction

In problems of high complexity in mathematics and physics it is often the case that a solution can only be generated as a finite number of terms of a perturbation series at certain special points. Under these circumstances, we ask what is the optimal strategy to approximate the un- derlying function, and if optimality cannot be achieved in practice, what are the most efficient near-optimal methods? As a mathematical question this is related to, but distinct from, conven- tional approximation theory. We formulate this mathematical problem as a problem of inverse approximation theory: here the approximation is fixed, in the form of a number n of terms of a series, and the underlying function F is to be reconstructed as accurately as possible in the large

1 n limit. For this purpose, again unlike in approximation theory, the simplicity of the functions and operators used in the reconstruction plays a secondary role. On a practical level, this reconstruction question has been encountered in many problems in the literature, and dealt with in various problem-specific ways [42,8, 46, 78, 14, 16]. However, an underlying rigorous mathematical foundation and theory seem to be lacking. We prove an optimality theorem (Theorem2), which constructs the best approximant (based on partial Maclaurin sums) to F in generic classes of functions analytic on a common Riemann surface Ω, and sharing a common rate of growth. Optimality is achieved with the uniformization map of Ω, which also provides rigorous numerical access to the entire Riemann surface. Theorem 10 constructs the explicit uniformization map for the Riemann surface in Borel plane for the tronquée solutions to the Painlevé eq