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Nuttall's theorem with analytic weights on algebraic S-contours
Given a function holomorphic at infinity, the -th diagonal Pad\'e
approximant to , denoted by , is a rational function of type
that has the highest order of contact with at infinity. Nuttall's
theorem provides an asymptotic formula for the error of approximation
in the case where is the Cauchy integral of a smooth density
with respect to the arcsine distribution on [-1,1]. In this note, Nuttall's
theorem is extended to Cauchy integrals of analytic densities on the so-called
algebraic S-contours (in the sense of Nuttall and Stahl)
Symmetric Contours and Convergent Interpolation
The essence of Stahl-Gonchar-Rakhmanov theory of symmetric contours as
applied to the multipoint Pad\'e approximants is the fact that given a germ of
an algebraic function and a sequence of rational interpolants with free poles
of the germ, if there exists a contour that is "symmetric" with respect to the
interpolation scheme, does not separate the plane, and in the complement of
which the germ has a single-valued continuation with non-identically zero jump
across the contour, then the interpolants converge to that continuation in
logarithmic capacity in the complement of the contour. The existence of such a
contour is not guaranteed. In this work we do construct a class of pairs
interpolation scheme/symmetric contour with the help of hyperelliptic Riemann
surfaces (following the ideas of Nuttall \& Singh and Baratchart \& the author.
We consider rational interpolants with free poles of Cauchy transforms of
non-vanishing complex densities on such contours under mild smoothness
assumptions on the density. We utilize -extension of the
Riemann-Hilbert technique to obtain formulae of strong asymptotics for the
error of interpolation
Strong Asymptotics of Hermite-Pad\'e Approximants for Angelesco Systems
In this work type II Hermite-Pad\'e approximants for a vector of Cauchy
transforms of smooth Jacobi-type densities are considered. It is assumed that
densities are supported on mutually disjoint intervals (an Angelesco system
with complex weights). The formulae of strong asymptotics are derived for any
ray sequence of multi-indices.Comment: 40 page
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