Section: New Results

Asymptotics of Rational Approximants

Participant : Laurent Baratchart.

This is joint work with M. Yattselev (IUPUI).

We studied best rational approximants in the sup norm to an analytic function f on compact set K of the analyticity domain Ω with connected complement. We showed that if the function can be continued analytically except over a set of logarithmic capacity zero comprising at most finitely many branchpoints, then the n-th root of the approximation error converges as n goes large to e-2/C, with C the minimal Green capacity in K of a compact set E outside of which f is single valued. Moreover, if C>0, the normalized counting measure of the poles converges to the Green equilibrium distribution on E. We are currently considering the case of infinitely many branchpoints so as to get a somewhat final result on weak asymptotics in rational approximation to functions with polar singular set.

The proof rests on a blend of AAK-theory and potential theory.