BEGIN:VCALENDAR VERSION:2.0 PRODID:-//Talks.cam//talks.cam.ac.uk// X-WR-CALNAME:Talks.cam BEGIN:VEVENT SUMMARY:Optimal reconstruction of functions from their truncated power ser ies at a point - Ovidiu Costin (Ohio State University) DTSTART:20210401T150000Z DTEND:20210401T160000Z UID:TALK158455@talks.cam.ac.uk CONTACT:INI IT DESCRIPTION:

I will speak about the question of the m athematically

optimal reconstruction of a function from a finite n umber of terms of its power

series at a point\, and on aditional d ata such as: as domain of analyticity\,

bounds or others.

\;

Aside from its intrinsic mathematical i nterest\, this

question is important in a variety of applications in mathematics and physics

such as the practical computation of th e Painleve transcendents\, which I will

use as an example\, and th e reconstruction of functions from resurgent

perturbative series i n models of quantum field theory and string theory. Given

a class of functions which have a common Riemann surface and a common type of boun ds

on it\, we show that the optimal procedure stems from the unifo rmization

theorem. A priori Riemann surface information and bounds exist for the Borel

transform of asymptotic expansions in wide cl asses of mathematical problems

such as meromorphic systems of line ar or nonlinear ODEs\, classes of PDEs and

many others\, \; kn own\, 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< br>
existing (generally ad-hoc) ones\, both theoretically and in their effective

numerical application\, which I will illustrate. The co mparison with Pade approximants

is especially interesting.

\;

When more specific information exists\, such as the nature

of the singularities of the functions of inter est\, we found methods based on

convolution operators to eliminate these singularities. The type of

singularities is known for resur gent functions coming from many problems in

analysis. With this ad dition\, the accuracy is improved substantially with

respect to th e optimal accuracy which would be possible in full generality.

\;

Work in collaboration with G. Dunne\, U. Co nn.

LOCATION:Seminar Room 2\, Newton Institute END:VEVENT END:VCALENDAR