Sharp asymptotics of the Lp approximation error for interpolation on block partitions |
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Authors: | Yuliya Babenko Tatyana Leskevich Jean-Marie Mirebeau |
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Affiliation: | 1.Department of Mathematics and Statistics,Kennesaw State University,Kennesaw,USA;2.Department of Mathematical Analysis and Theory of Functions,Dnepropetrovsk National University,Dnepropetrovsk,Ukraine;3.Laboratoire Jacques-Louis Lions,UPMC Univ Paris 06, UMR 7598,Paris,France;4.Laboratoire Jacques-Louis Lions,CNRS, UMR 7598,Paris,France |
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Abstract: | Adaptive approximation (or interpolation) takes into account local variations in the behavior of the given function, adjusts the approximant depending on it, and hence yields the smaller error of approximation. The question of constructing optimal approximating spline for each function proved to be very hard. In fact, no polynomial time algorithm of adaptive spline approximation can be designed and no exact formula for the optimal error of approximation can be given. Therefore, the next natural question would be to study the asymptotic behavior of the error and construct asymptotically optimal sequences of partitions. In this paper we provide sharp asymptotic estimates for the error of interpolation by splines on block partitions in mathbbRd{mathbb{R}^d} . We consider various projection operators to define the interpolant and provide the analysis of the exact constant in the asymptotics as well as its explicit form in certain cases. |
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