Optimal reconstruction of functions from their truncated power series at a point
1 hour 2 mins,
115.21 MB,
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| Description: |
Costin, O
Thursday 1st April 2021 - 16:00 to 17:00 |
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| Created: | 2021-04-06 09:03 |
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| Collection: | Applicable resurgent asymptotics: towards a universal theory |
| Publisher: | Isaac Newton Institute |
| Copyright: | Costin, O |
| Language: | eng (English) |
| Distribution: |
World
(downloadable)
|
| Explicit content: | No |
| Aspect Ratio: | 16:9 |
| Screencast: | No |
| Bumper: | UCS Default |
| Trailer: | UCS Default |
| Abstract: | 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. |
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