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Nonlinear Nonnested Spline Approximation

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Abstract

Nonlinear approximation from regular piecewise polynomials (splines) supported on rings in \(\mathbb {R}^2\) is studied. By definition, a ring is a set in \(\mathbb {R}^2\) obtained by subtracting a compact convex set with polygonal boundary from another such a set, but without creating uncontrollably narrow elongated subregions. Nested structure of the rings is not assumed; however, uniform boundedness of the eccentricities of the underlying convex sets is required. It is also assumed that the splines have maximum smoothness. Bernstein type inequalities for this sort of splines are proved that allow us to establish sharp inverse estimates in terms of Besov spaces.

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Acknowledgements

We would like to give credit to Peter Petrov (Sofia University) with whom the second author discussed the theme of this article some years ago.

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Authors and Affiliations

  1. Department of Mathematics, Karlstad University, 651 88, Karlstad, Sweden

    Martin Lind

  2. Department of Mathematics, University of South Carolina, Columbia, SC, 29208, USA

    Pencho Petrushev

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  1. Martin Lind
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  2. Pencho Petrushev
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Correspondence to Pencho Petrushev.

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Communicated by Ronald A. DeVore.

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Lind, M., Petrushev, P. Nonlinear Nonnested Spline Approximation. Constr Approx 45, 143–191 (2017). https://doi.org/10.1007/s00365-016-9361-3

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  • DOI: https://doi.org/10.1007/s00365-016-9361-3

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