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 approximants
of degree 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
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 rational approximation to
some badly approximable functions (Blaschke products) and showed
equivalence, up to a constant, of best and 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] .