A Borel transform method for locating singularities of Taylor and Fourier series

Uriel Frisch (Observatoire de Nice)

Observatoire de Nice

Given a Taylor series  with a finite radius ofconvergence, its Borel transform defines an entire function. A theorem of
P\'olya relates the large distance behavior of the Borel transform in different directions to singularities of the original
function. With the help of the new asymptotic interpolation method of van der Hoeven, we show that from the knowledge
of a large number  of Taylor coefficients we can identify precisely the location of such singularities, as well as their type
when they are isolated. There is no risk of getting artefacts with this method, which also gives us access to some of the singularities
beyond the convergence disk. The  method can also be applied to Fourier series of analytic periodic functions and is here tested on
various instances constructed from solutions to the Burgers equation. Large  precision on scaling exponents (up to twenty
accurate digits) can be achieved. [based on the paper by W. Pauls and U. Frisch  nlin.CD/0609025 ]