Maxim Yattselev
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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