Lower bounds for quadrature and recovery in L2

David Krieg (JKU, Linz)

Jan 21. 2022, 16:30 — 17:10

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 discuss new lower bounds. In particular we show 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 give new lower bounds for the integration problem (quadrature), again for rather classical Sobolev spaces of periodic univariate functions.

Further Information
ESI Boltzmann Lecture Hall
Associated Event:
Optimal Point Configurations on Manifolds (Workshop)
Christine Bachoc (U Bordeaux)
Henry Cohn (Microsoft, Redmond)
Peter Grabner (TU Graz)
Douglas Hardin (Vanderbilt U, Nashville)
Edward Saff (Vanderbilt U, Nashville)
Achill Schürmann (U of Rostock)
Robert Womersley (UNSW, Sydney)