On \(L_2\)-approximation in Hilbert spaces using function values

Abstract

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 \(d>2s+1\).