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Approximation by Chebyshevian Bernstein Operators versus Convergence of Dimension Elevation

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Abstract

On a closed bounded interval, consider a nested sequence of Extended Chebyshev spaces possessing Bernstein bases. This situation automatically generates an infinite dimension elevation algorithm transforming control polygons of any given level into control polygons of the next level. The convergence of these infinite sequences of polygons towards the corresponding curves is a classical issue in computer-aided geometric design. Moreover, according to recent work proving the existence of Bernstein-type operators in such Extended Chebyshev spaces, this nested sequence is automatically associated with an infinite sequence of Bernstein operators which all reproduce the same two-dimensional space. Whether or not this sequence of operators converges towards the identity on the space of all continuous functions is a natural issue in approximation theory. In the present article, we prove that the two issues are actually equivalent. Not only is this result interesting on the theoretical side, but it also has practical implications. For instance, it provides us with a Korovkin-type theorem of convergence of any infinite dimension elevation algorithm. It also enables us to tackle the question of convergence of the dimension elevation algorithm for any nested sequence obtained by repeated integration of the kernel of a given linear differential operator with constant coefficients.

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Acknowledgments

We would like to warmly thank the anonymous referees whose comments, questions and suggestions helped us improve the paper.

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Authors and Affiliations

  1. King Abdullah University of Science and Technology, Thuwal, Saudi Arabia

    Rachid Ait-Haddou

  2. Laboratoire Jean Kuntzmann, CNRS, UMR 5224, Université Grenoble Alpes, BP 53, 38041, Grenoble Cedex 9, France

    Marie-Laurence Mazure

Authors
  1. Rachid Ait-Haddou
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  2. Marie-Laurence Mazure
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Correspondence to Marie-Laurence Mazure.

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Communicated by : Wolfgang Dahmen.

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Ait-Haddou, R., Mazure, ML. Approximation by Chebyshevian Bernstein Operators versus Convergence of Dimension Elevation. Constr Approx 43, 425–461 (2016). https://doi.org/10.1007/s00365-016-9331-9

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  • DOI: https://doi.org/10.1007/s00365-016-9331-9

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