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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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Submitted on : Monday, August 2, 2010 - 11:33:46 PM
Last modification on : Thursday, January 20, 2022 - 4:12:21 PM
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  • HAL Id : inria-00119160, version 2



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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