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 L² norm optimization process and the results are validated by comparison with the unconstrained L² 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.