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 on compact set of the analyticity domain 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 -th root of the approximation error converges as goes large to , with the minimal Green capacity in of a compact set outside of which is single valued. Moreover, if , the normalized counting measure of the poles converges to the Green equilibrium distribution on . 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.