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Analysis Mathematica

, Volume 43, Issue 2, pp 133–191 | Cite as

Optimal sampling recovery of mixed order Sobolev embeddings via discrete Littlewood–Paley type characterizations

  • G. Byrenheid
  • T. Ullrich
Article

Abstract

In this paper we consider the L q -approximation of multivariate periodic functions f with L q -bounded mixed derivative (difference). The (possibly non-linear) reconstruction algorithm is supposed to recover the function from function values, sampled on a discrete set of n sampling nodes. The general performance is measured in terms of (non-)linear sampling widths ϱ n . We conduct a systematic analysis of Smolyak type interpolation algorithms in the framework of Besov–Lizorkin–Triebel spaces of dominating mixed smoothness based on specifically tailored discrete Littlewood–Paley type characterizations. As a consequence, we provide sharp upper bounds for the asymptotic order of the (non-)linear sampling widths in various situations and close some gaps in the existing literature. For example, in case 2 ≤ p < q < ∞ and r > 1/p the linear sampling widths ϱ n lin (S p r W(T d ), L q (T d )) and ϱ n lin (S p,∞ r B(T d ), L q (T d )) show the asymptotic behavior of the corresponding Gelfand n-widths, whereas in case 1 < p < q ≤ 2 and r > 1/p the linear sampling widths match the corresponding linear widths. In the mentioned cases linear Smolyak interpolation based on univariate classical trigonometric interpolation turns out to be optimal.

Key words and phrases

sampling recovery sparse grid sampling representation Besov–Triebel–Lizorkin space of mixed smoothness Smolyak algorithm Gelfand n-width linear width 

Mathematics Subject Classification

42A10 42A15 41A46 41A58 41A63 41A17 41A25 42B25 42B35 

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Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2017

Authors and Affiliations

  1. 1.Institute for Numerical SimulationHausdorff Center for MathematicsBonnGermany

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