Padé approximants and noise: A case of geometric series

Abstract

We study effects, on Padé approximants (PA), of a random noise added to coefficients of a power series. Numerical experiments performed during the last 30 years have shown that such noise results in the appearance of, the so-called, Froissart doublets (FD) — pairs of zeros and poles separated by a distance of a scale of the noise. For the first time we can prove that this effect actually takes place for geometric series. We can also show what happens with noise-induced poles or zeros that do not form FD. We construct a polynomial that has as its roots mean positions of FD. Coefficients of this polynomial are expressed by random perturbations of coefficients of the series. We study numerically this polynomial for low orders to show where (in these orders) FD coalesce. Numerical study of this polynomial, to see a behaviour of FD for large orders, is much easier than a study of the entire PA. We also express a deviation of a pole of PAs approximating the true pole of 1 (1 − z) , from 1, by perturbations of coefficients and show numerically that when order of PAs increases, this deviation drops down.