Meromorphic Approximants to Complex Cauchy Transforms with Polar Singularities

Details Meromorphic Approximants to Complex Cauchy Transforms with Polar Singularities Meromorphic Approximants to Complex Cauchy Transforms with Polar Singularities

2008-06-28

arxiv.org/abs/0806.4681v1

Mat. Sb., 200(9), 3-40, 2009

Mathematics

Classical Analysis and ODEs

39 pages, 4 figures

Scientific paper

We study AAK-type meromorphic approximants to functions $F$, where $F$ is a sum of a rational function $R$ and a Cauchy transform of a complex measure $\lambda$ with compact regular support included in $(-1,1)$, whose argument has bounded variation on the support. The approximation is understood in $L^p$-norm of the unit circle, $p\geq2$. We obtain that the counting measures of poles of the approximants converge to the Green equilibrium distribution on the support of $\lambda$ relative to the unit disk, that the approximants themselves converge in capacity to $F$, 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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