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 $\Omega$ 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 $ℂ\setminus K$ 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.