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 DũngEmail author


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 \(r_{n}({U^{r}_{p}}, L_{q})\).


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 



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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