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Acta Mathematica Vietnamica

, Volume 43, Issue 1, pp 83–110 | Cite as

B-Spline Quasi-Interpolation Sampling Representation and Sampling Recovery in Sobolev Spaces of Mixed Smoothness

  • Dinh Dũng
Article

Abstract

We proved direct and inverse theorems on B-spline quasi-interpolation sampling representation with a Littlewood-Paley-type norm equivalence in Sobolev spaces \({W^{r}_{p}}\) of mixed smoothness r. Based on this representation, we established estimates of the approximation error of recovery in L q -norm of functions from the unit ball \({U^{r}_{p}}\) in the spaces \({W^{r}_{p}}\) by linear sampling algorithms and the asymptotic optimality of these sampling algorithms in terms of Smolyak sampling width \({r^{s}_{n}}({U^{r}_{p}}, L_{q})\) and sampling width \(r_{n}({U^{r}_{p}}, L_{q})\).

Keywords

Sampling width Linear sampling algorithms Smolyak grids Sobolev spaces of mixed smoothness B-spline quasi-interpolation sampling representations Littlewood-Paley-type theorem 

Mathematics Subject Classification (2010)

41A15 41A05 41A25 41A58 41A63 

Notes

Acknowledgements

This work is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant No. 102.01-2017.05. A part of this work was done when the author was working as a research professor at the Vietnam Institute for Advanced Study in Mathematics (VIASM). He would like to thank the VIASM for providing a fruitful research environment and working condition. The author would like to thank Glenn Byrenheid and Tino Ullrich for giving opportunity to read the manuscript [7]. He thanks Glenn Byrenheid, Vladimir Temlyakov, and Tino Ullrich for useful discussions.

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Copyright information

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2017

Authors and Affiliations

  1. 1.Information Technology InstituteVietnam National UniversityHanoiVietnam

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