Section: New Results

Rational and meromorphic approximation

Participant : Laurent Baratchart.

This work has been done in collaboration with Herbert Stahl (TFH Berlin) and Maxim Yattselev (Univ. Oregon at Eugene, USA).

We completed and published this year the proof of an important result in approximation theory, namely the counting measure of poles of best $H^2$ approximants (more generally: of critical points) of degree $n$ to a function analytically continuable, except over finitely many branchpoints lying outside the unit disk, converges to the Green equilibrium distribution of the compact set of minimal Green capacity outside of which the function is single valued [14] . The proof requires showing existence and uniqueness of a compact set of minimal weighted logarithmic capacity in a field, outside of which the function is single-valued. Structure of this contour, along with error estimates, also come out of the proof. The result is in fact valid for functions that are Cauchy integrals of Dini-smooth functions on such a contour. We rely in addition on asymptotic interpolation estimates from [63] .

This result warrants source recovery techniques used in section 6.1.1 .

We also studied partial realizations, or equivalently Padé approximants to transfer functions with branchpoints. Identification techniques based on partial realizations of a stable infinite-dimensional transfer function are known to often provide unstable models, but the question as to whether this is due to noise or to intrinsic instability was not clear. In the case of 4 branchpoints, expressing the computation of Padé approximants in terms of the solution to a Riemann-Hilbert problem on the Riemann surface of the function, we proved that the pole behaviour generically shows deterministic chaos [49] .