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

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ARA - Applicable resurgent asymptotics: towards a universal theory

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, and on aditional data such as: as domain of analyticity,

bounds or others.


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 resurgent

perturbative series 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 and bounds exist for the Borel

transform of asymptotic expansions in wide classes of mathematical problems

such as meromorphic systems of linear or nonlinear ODEs, classes of PDEs and

many others,  known, by mathematical

theorems,  to be resurgent.  I will also discuss some (apparently) new

uniformization methods and maps. Explicit uniformization in Borel plane is

possible for all linear or nonlinear second order meromorphic ODEs.


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. The type of

singularities is known for resurgent functions coming from many problems in

analysis. 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.

This talk is part of the Isaac Newton Institute Seminar Series series.

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