Multipoint Padé Approximants to Complex Cauchy Transforms with Polar Singularities

Abstract : We study diagonal multipoint Padé approximants to functions of the form \[F(z) = \int\frac{d\mes(t)}{z-t}+R(z),\] where $R$ is a rational function and $\mes$ is a complex measure with compact regular support included in $\R$, whose argument has bounded variation on the support. Assuming that interpolation sets are such that their normalized counting measures converge sufficiently fast in the weak-star sense to some conjugate-symmetric distribution $\sigma$, we show that the counting measures of poles of the approximants converge to $\widehat\sigma$, the balayage of $\sigma$ onto the support of $\mes$, in the weak$^*$ sense, that the approximants themselves converge in capacity to $F$ outside the support of $\mes$, and that the poles of $R$ attract at least as many poles of the approximants as their multiplicity and not much more.
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Journal of Approximation Theory, Elsevier, 2009, 156 (2), pp.187-211
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Laurent Baratchart, Maxim Yattselev. Multipoint Padé Approximants to Complex Cauchy Transforms with Polar Singularities. Journal of Approximation Theory, Elsevier, 2009, 156 (2), pp.187-211. 〈inria-00119160v2〉

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