Lower bounds for integration and recovery in L2

Function values are, in some sense, “almost as good” as general linear information for L2-approximation (optimal recovery, data assimilation) of functions from a reproducing kernel Hilbert space. This was recently proved by new upper bounds on the sampling numbers under the assumption that the singular values of the embedding of this Hilbert space into L2 are square-summable. Here we mainly prove new lower bounds. In particular we prove that the sampling numbers behave worse than the approximation numbers for Sobolev spaces with small smoothness. Hence there can be a logarithmic gap also in the case where the singular numbers of the embedding are square-summable. We first prove new lower bounds for the integration problem, again for rather classical Sobolev spaces of periodic univariate functions. Institut für Analysis, Johannes Kepler Universität Linz, Altenbergerstrasse 69, 4040 Linz, Austria, Email: aicke.hinrichs@jku.at, david.krieg@jku.at; the research of these authors is supported by the Austrian Science Fund (FWF) Project F5513-N26, which is a part of the Special Research Program Quasi-Monte Carlo Methods: Theory and Applications. Mathematisches Institut, FSU Jena, Ernst-Abbe-Platz 2, 07740 Jena, Germany, Email: erich.novak@uni-jena.de. Dept. of Mathematics FNSPE, Czech Technical University in Prague, Trojanova 13, 12000 Prague, Czech Republic, Email: jan.vybiral@fjfi.cvut.cz; the research of this author was supported by the European Regional Development Fund-Project “Center for Advanced Applied Science” (No. CZ.02.1.01/0.0/0.0/16 019/0000778).

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