Optimal reconstruction of functions from their truncated power series at a point

I will speak about the question of the mathematically optimal reconstruction of a function from a finite number of terms of its power series at a point. Aside from its intrinsic mathematical interest, this question is important in a variety of applications in mathematics and physics such as the practical computation of the Painleve transcendents, which I will use as an example, and the reconstruction of functions from perturbative expansions (usually divergent but resurgent, in particular generalized Borel summable) in models of quantum field theory and string theory. Given a class of functions which have a common Riemann surface and a common type of bounds on it, we show that the optimal procedure stems from the uniformization theorem. A priori Riemann surface information exists for the Borel transform of asymptotic expansions in wide classes of mathematical problems, including meromorphic systems of linear or nonlinear ODEs, classes of PDEs among many others. I will also discuss some possibly new uniformization methods and maps.
This optimal procedure is dramatically superior to the existing (generally ad-hoc) ones, both theoretically and in their effective numerical application, which I will illustrate. The comparison with Pade approximants is especially interesting.
When more specific information exists, such as the nature of the singularities of the functions of interest, we found methods based on convolution operators to eliminate these singularities. With this addition, the accuracy is improved substantially with respect to the optimal accuracy which would be possible in full generality.
Work in collaboration with G. Dunne, U. Conn.