Davydov, O. and Petrushev, P.
(2003)
*Nonlinear approximation from differentiable piecewise polynomials.*
SIAM Journal on Mathematical Analysis, 35 (3).
pp. 708-758.
ISSN 0036-1410

## Abstract

We study nonlinear $n$-term approximation in $L_p({mathbb R}^2)$ ($0<pleinfty$) from hierarchical sequences of stable local bases consisting of differentiable (i.e., $C^r$ with $rge1$) piecewise polynomials (splines). We construct such sequences of bases over multilevel nested triangulations of ${mathbb R}^2$, which allow arbitrarily sharp angles. To quantize nonlinear n-term spline approximation, we introduce and explore a collection of smoothness spaces (B-spaces). We utilize the B-spaces to prove companion Jackson and Bernstein estimates and then characterize the rates of approximation by interpolation. Even when applied on uniform triangulations with well-known families of basis functions such as box splines, these results give a more complete characterization of the approximation rates than the existing ones involving Besov spaces. Our results can easily be extended to properly defined multilevel triangulations in ${mathbb R}^d$, d>2.

Item type: | Article |
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ID code: | 2171 |

Keywords: | nonlinear approximation, Jackson and Bernstein estimates, multivariate splines, multilevel nested triangulations, multilevel bases, stable local spline bases, Mathematics, Computational Mathematics, Analysis, Applied Mathematics |

Subjects: | Science > Mathematics |

Department: | Faculty of Science > Mathematics and Statistics |

Depositing user: | Strathprints Administrator |

Date Deposited: | 04 Jan 2007 |

Last modified: | 21 May 2015 08:32 |

URI: | http://strathprints.strath.ac.uk/id/eprint/2171 |

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