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

2016/03/15 by Byrenheid, Glenn, Ullrich, Tino
#41A17 #41A25 #41A46 #41A58 #41A63 #42A10 #42A15 #42B25 #42B35 #FOS: Mathematics #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.1603.04809

Abstract

In this paper we consider the L_q-approximation of multivariate periodic functions f with L_p-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 \varrho_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≤ p1/p the linear sampling widths \varrho_n^lin(S^r_pW(\mathbbT^d),L_q(\mathbbT^d)) and \varrho^lin_n(S^r_p,∞B(\mathbbT^d),L_q(\mathbbT^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.

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