Section: New Results
Rational and meromorphic approximation
Participants : Laurent Baratchart, Herbert Stahl [TFH Berlin] , Maxim Yattselev.
We demonstrated in a recent past, under mild smoothness assumptions, the possibility of convergent rational interpolation to Cauchy integrals of complex measures on analytic Jordan arcs and their strong asymptotics  ,  . Subsequently, we started investigating the case of Cauchy integrals on so-called symmetric contours for the logarithmic potential. These correspond to functions with more than two branched singularities, like those arising in the slicing method for source recovery in a sphere when there is more than one source (see section 6.2.2 ). Recently we obtained weak asymptotics in this case, and dwelled on them to elucidate the asymptotic of poles of best meromorphic approximants of given degree to a function with branched singularities on a curve encompassing them. Namely, the counting measure of the poles converges weak-star, when the degree goes large, to the Green equilibrium distribution of the set with minimum Green capacity inside the curve, outside of which the function is single-valued. The technical core of this contribution is an existence and uniqueness result, along with a differential characterization, of the compact of minimum weighted capacity outside of which the function is single-valued  . This teams up with results from  to produce the results.
We presently study strong asymptotics, limiting ourselves at present to a threefold geometry, and to the case of Padé approximants (interpolation at a single point with high order). The result is that uniform convergence can only take place if the weights of the branches of the threefold with respect to the equilibrium distribution are rational. If they are rationally dependent, a spurious pole clusters to certain curves within the domain of analyticity, and if they are rationally independent, exactly one pole exhibits chaotic behaviour in the complex plane. Moreover, we have shown that the chaotic situation is generic, in a measure theoretic sense, with respect to the location of branchpoints. This generalizes and sharpens results of Suetin for Cauchy integrals on disconnected pieces of a smooth symmetric contour. It is the first time that a branched contour is analyzed with respect to general densities. A paper is being written to report these results.