Laurent Baratchart
Minimal symmetric Darlington synthesis.
Wed, 05/04/2011 - 15:01 — SomNambulAuthors: Laurent Baratchart, Per Enqvist, Andrea Gombani, Martine Olivi
Subjects: Optimization and Control
AbstractWe consider the symmetric Darlington synthesis of a p x p rational symmetric
Schur function S with the constraint that the extension is of size 2p x 2p.
Under the assumption that S is strictly contractive in at least one point of
the imaginary axis, we determine the minimal McMillan degree of the extension.
In particular, we show that it is generically given by the number of zeros of
odd multiplicity of I-SS*. A constructive characterization of all such
extensions is provided in terms of a symmetric realization of S and of the
outer spectral factor of I-SS*.Convergent Interpolation to Cauchy Integrals over Analytic Arcs with Jacobi-Type Weights.
Fri, 11/20/2009 - 16:01 — SomNambulAuthors: Laurent Baratchart, Maxim Yattselev
Subjects: Classical Analysis and ODEs
AbstractWe design convergent multipoint Pade interpolation schemes to Cauchy
transforms of non-vanishing complex densities with respect to Jacobi-type
weights on analytic arcs, under mild smoothness assumptions on the density. We
rely on our earlier work for the choice of the interpolation points, and dwell
on the Riemann-Hilbert approach to asymptotics of orthogonal polynomials
introduced by Kuijlaars, McLaughlin, Van Assche, and Vanlessen in the case of a
segment.Asymptotic Uniqueness of Best Rational Approximants to Complex Cauchy Transforms in ${L}^2$ of the Circle.
Thu, 09/03/2009 - 17:38 — SomNambulAuthors: Laurent Baratchart, Maxim Yattselev
Subjects: Classical Analysis and ODEs
AbstractFor all n large enough, we show uniqueness of a critical point in best
rational approximation of degree n, in the L^2-sense on the unit circle, to
functions f, where f is a sum of a Cauchy transform of a complex measure \mu
supported on a real interval included in (-1,1), whose Radon-Nikodym derivative
with respect to the arcsine distribution on its support is Dini-continuous,
non-vanishing and with and argument of bounded variation, and of a rational
function with no poles on the support of \mu.
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