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The Genuine Bernstein–Durrmeyer Operators and Quasi-Interpolants

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Abstract

In this paper we consider the so-called genuine Bernstein–Durrmeyer operators and define corresponding quasi-interpolants of order \({r \in \mathbb{N}_0}\) in terms of certain differential operators. These quasi-interpolants preserve all polynomials of degree at most r + 1. We analyse the eigenstructure of the differential operators and the quasi-interpolants and prove as main results an error estimate of Jackson–Favard type for sufficiently smooth functions and an upper bound for the error of approximation in the sup-norm in terms of an appropriate K-functional.

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Authors and Affiliations

  1. Faculty of Mathematics and Natural Sciences, University of Wuppertal, Gaußstraße 20, 42119, Wuppertal, Germany

    Margareta Heilmann & Martin Wagner

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  1. Margareta Heilmann
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  2. Martin Wagner
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Correspondence to Margareta Heilmann.

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Dedicated to the memory of Werner Haußmann.

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Heilmann, M., Wagner, M. The Genuine Bernstein–Durrmeyer Operators and Quasi-Interpolants. Results. Math. 62, 319–335 (2012). https://doi.org/10.1007/s00025-012-0247-9

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  • DOI: https://doi.org/10.1007/s00025-012-0247-9

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