Foundations of Computational Mathematics

, Volume 16, Issue 5, pp 1193–1240 | Cite as

Sampling and Cubature on Sparse Grids Based on a B-spline Quasi-Interpolation

  • Dinh Dũng


Let \(X_n = \{x^j\}_{j=1}^n\) be a set of n points in the d-cube \({\mathbb {I}}^d:=[0,1]^d\), and \(\Phi _n = \{\varphi _j\}_{j =1}^n\) a family of n functions on \({\mathbb {I}}^d\). We consider the approximate recovery of functions f on \({{\mathbb {I}}}^d\) from the sampled values \(f(x^1), \ldots , f(x^n)\), by the linear sampling algorithm \( L_n(X_n,\Phi _n,f) := \sum _{j=1}^n f(x^j)\varphi _j. \) The error of sampling recovery is measured in the norm of the space \(L_q({\mathbb {I}}^d)\)-norm or the energy quasi-norm of the isotropic Sobolev space \(W^\gamma _q({\mathbb {I}}^d)\) for \(1 < q < \infty \) and \(\gamma > 0\). Functions f to be recovered are from the unit ball in Besov-type spaces of an anisotropic smoothness, in particular, spaces \(B^{\alpha ,\beta }_{p,\theta }\) of a “hybrid” of mixed smoothness \(\alpha > 0\) and isotropic smoothness \(\beta \in {\mathbb {R}}\), and spaces \(B^a_{p,\theta }\) of a nonuniform mixed smoothness \(a \in {\mathbb {R}}^d_+\). We constructed asymptotically optimal linear sampling algorithms \(L_n(X_n^*,\Phi _n^*,\cdot )\) on special sparse grids \(X_n^*\) and a family \(\Phi _n^*\) of linear combinations of integer or half integer translated dilations of tensor products of B-splines. We computed the asymptotic order of the error of the optimal recovery. This construction is based on B-spline quasi-interpolation representations of functions in \(B^{\alpha ,\beta }_{p,\theta }\) and \(B^a_{p,\theta }\). As consequences, we obtained the asymptotic order of optimal cubature formulas for numerical integration of functions from the unit ball of these Besov-type spaces.


Linear sampling algorithms Optimal sampling recovery  Cubature formulas Optimal cubature Sparse grids Besov-type spaces of anisotropic smoothness B-spline quasi-interpolation representations 

Mathematics Subject Classification

41A15 41A05 41A25 41A58 41A63 



This work is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant No. 102.01-2014.02. 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 specially thank Dr. Tino Ullrich and Glenn Byrenheid for their valuable remarks and suggestions. He thanks the referees for constructive remarks, comments and suggestions which certainly improved the presentation of the paper.


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Copyright information

© SFoCM 2015

Authors and Affiliations

  1. 1.Information Technology InstituteVietnam National University, HanoiHanoiVietnam

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