## Section: New Results

### Schur rational approximation

Participants : Laurent Baratchart, Stanislas Kupin [ Univ. Bordeaux 1 ] .

Passive devices play an important role in many application areas:
telecommunication, chemical process
control, economy, biomedical processes. Network simulation software
packages (as ADS or SPICE) require
passive models for their components. However, identifying a passive model
from band limited frequency data
is still an open and challenging problem. Schur rational approximation is
a new way to approach this problem and has been the subject of [80] .
In this
work, a parametrization of all strictly
Schur rational functions of degree n is constructed from a multipoint
Schur algorithm, the parameters being
both the interpolation values and interpolation points. Examples are
computed by an L^{2} norm optimization
process and the results are validated by comparison with the unconstrained
L^{2} rational approximation. Over the last two years,
the results of [76]
on the hyperbolic convergence of the classical Schur
algorithm were generalized to the
case of the multipoint Schur algorithm, which is a more delicate situation
because we allow for the interpolation points to approach the boundary
circle. Orthogonal rational functions and a recent generalization of
Geronimus
theorem were used [79] , combined with Hilbertian techniques from
reproducing kernel spaces and new quantitative versions of the Beurling
theorem, to obtain an analog of the Szegö theorem where the interpolation
points
tend to the boundary, provided the approximated function is continuous and
less than 1 in a neighborhood of
the accumulation set of the interpolation points. This generalized the
results in [60]
and was of novel type since
the n-th orthogonal rational function inverts the Szegö function modulo
the Poisson kernel. This yields a rather unexpected theorem on the
behaviour of certain orthogonal polynomials with varying weight. This year
we obtained new bounds for orthogonal rational functions, based on
-estimates for the squared modulus of the Szegö
function, that yield new results even in the case of polynomials since
they yield information even in cases where the measure has vanishing
density provided it is Sobolev-W^{1-1/p, p} -smooth on the circle for
some p>1 .
An article is being written that summarizes these
results.

The research has been pursued on a major open issue, namely how to choose the interpolation points with respect to the approximated Schur function so as to yield the best convergence possible. We also started analyzing the consequences of our work for the representation of certain non-stationary stochastic processes. In this connection the case of vanishing densities and singular components is under investigation.