On the convergence of generalized Padé approximants

AbstractLetf be an analytic function with all its singularities in a compact set $$E_f \subset \bar C$$ of (logarithmic) capacity zero. The function may have branch points. The convergence of generalized (multipoint) Padé approximants to this type of function is investigated. For appropriately selected schemes of interpolation points, it is shown that close-to-diagonal sequences of generalized Padé approximants converge in capacity tof in a certain domain that can be characterized by the property of the minimal condenser capacity. Using a pole elimination procedure, another set of rational approximants tof is derived from the considered generalized Padé approximants. These new approximants converge uniformly on a given continuum $$V \subset \bar C\backslash E_f$$ with a rate of convergence that has been conjectured to be best possible. The continuumV is assumed not to divide the complex plane.

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