Sharp asymptotics of the Lp approximation error for interpolation on block partitions

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 IR. 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. 1 ar X iv :1 10 1. 17 76 v1 [ m at h. N A ] 1 0 Ja n 20 11 Yuliya Babenko Department of Mathematics and Statistics Sam Houston State University Box 2206 Huntsville, TX, USA 77340-2206 Phone: 936.294.4884 Fax: 936.294.1882 Email: [email protected] Tatyana Leskevich Department of Mathematical Analysis Dnepropetrovsk National University pr. Gagarina, 72, Dnepropetrovsk, UKRAINE, 49050 Email: [email protected] Jean-Marie Mirebeau UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France Email: [email protected]