Section: New Results

Behavior of poles in meromorphic approximation

Participant : Laurent Baratchart.

We proved this year that if a function is holomorphic outside a disk of radius $r<1$ in the complex plane, then its best approximant on the unit circle, in the uniform norm, by a meromorphic function having at most $n$ poles in the unit disk has at most $m$ poles of modulus greater than $r$, where $m$ is independent of $n$. This is the first result on the behavior of singularities in meromorphic approximation to a function with 2-D singular set. We are currently working on analogs in a non-circular geometry and in rational rather than meromorphic approximation.