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 ]

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 ]