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Section: New Results

Rational and meromorphic approximation

Participants : Laurent Baratchart, Sylvain Chevillard.

This work has been done in collaboration with Herbert Stahl (Beuth-Hochsch.), Maxim Yattselev (Purdue Univ. at Indianapolis, USA), Tao Qian (Univ. Macao).

We published last year an important result in approximation theory, namely the counting measure of poles of best $H^2$ approximants 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 [6] (see also [21] ). This result warrants source recovery techniques used in Section  6.1.1 . We considered this year a similar problem for best uniform meromorphic approximants on the unit circle (so-called AAK approximants after Adamjan, Arov and Krein), in the case where the function may have poles and essential singularities. The technical difficulties are considerable, and though a line of attack has been adopted we presently struggle with the proof.

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. This year, we published a paper showing that, in the case of 4 branchpoints, the pole behavior generically has deterministic chaos to it [15] .

We also considered the issue of lower bounds in rational approximation. Prompted by renewed interest for linearizing techniques such as vector fitting in the identification community, we studied linearized errors in light of the topological approach in [51] , to find that, when properly normalized, they give rise to lower bounds in $L^2$ rational approximation. Moreover, these make contact with AAK theory which furnishes more, easily computable lower bounds. This is an interesting finding, for lower bounds are usually difficult to get in approximation and though quite helpful to get an appraisal of what can be hoped for in modeling. Dwelling on this, we established for the first time lower bounds in $L^2$ rational approximation to some badly $L^{\infty }$ approximable functions (Blaschke products) and showed equivalence, up to a constant, of best $L^2$ and $L^{\infty }$ approximation to functions with branchpoints (such as those appearing in inverse source problems for EEG, see Section  6.1.1 ). An article on this subject is currently submitted for publication in the Journal of Approximation Theory [29] .