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Convergent Interpolation to Cauchy Integrals over Analytic Arcs

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Abstract

We consider multipoint Padé approximation to Cauchy transforms of complex measures. We show that if the support of a measure is an analytic Jordan arc and if the measure itself is absolutely continuous with respect to the equilibrium distribution of that arc with Dini-smooth nonvanishing density, then the diagonal multipoint Padé approximants associated with appropriate interpolation schemes converge locally uniformly to the approximated Cauchy transform in the complement of the arc. This asymptotic behavior of Padé approximants is deduced from the analysis of underlying non-Hermitian orthogonal polynomials, for which we use classical properties of Hankel and Toeplitz operators on smooth curves. A construction of the appropriate interpolation schemes is explicit granted the parametrization of the arc.

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  1. INRIA, Project APICS, 2004 route des Lucioles, BP 93, 06902, Sophia-Antipolis, France

    Laurent Baratchart & Maxim Yattselev

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  1. Laurent Baratchart
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  2. Maxim Yattselev
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Correspondence to Maxim Yattselev.

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Communicated by Arieh Iserles.

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Baratchart, L., Yattselev, M. Convergent Interpolation to Cauchy Integrals over Analytic Arcs. Found Comput Math 9, 675–715 (2009). https://doi.org/10.1007/s10208-009-9042-8

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