On $L_2$-approximation in Hilbert spaces using function values

David Krieg, Mario Ullrich

Published 2019-05-07Version 1

We study $L_2$-approximation of functions from Hilbert spaces $H$ in which function evaluation is a continuous linear functional, using function values as information. Under certain assumptions on $H$, we prove that the $n$-th minimal worst-case error $e_n$ satisfies \[ e_n \,\lesssim\, a_{n/\log(n)}, \] where $a_n$ is the $n$-th minimal worst-case error for algorithms using arbitrary linear information, i.e., the $n$-th approximation number. Our result applies, in particular, to Sobolev spaces with dominating mixed smoothness $H=H^s_{\rm mix}(\mathbb{T}^d)$ with $s>1/2$ and we obtain \[ e_n \,\lesssim\, n^{-s} \log^{sd}(n). \] This improves upon previous bounds whenever $d>2s+1$.