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

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

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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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Correspondence to Dinh Dũng.

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Dũng, D. B-Spline Quasi-Interpolation Sampling Representation and Sampling Recovery in Sobolev Spaces of Mixed Smoothness. Acta Math Vietnam 43, 83–110 (2018). https://doi.org/10.1007/s40306-017-0230-3

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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