Tractability of $\mathbb{L}_2$-approximation in hybrid function spaces

Abstract

We consider multivariate $\mathbb{L}_2$-approximation in reproducing kernel Hilbert spaces which are tensor products of weighted Walsh spaces and weighted Korobov spaces. We study the minimal worst-case error $e^{\mathbb{L}_2-\mathrm{app},Λ}(N,d)$ of all algorithms that use $N$ information evaluations from the class $Λ$ in the $d$-dimensional case. The two classes $Λ$ considered in this paper are the class $Λ^{\rm all}$ consisting of all linear functionals and the class $Λ^{\rm std}$ consisting only of function evaluations. The focus lies on the dependence of $e^{\mathbb{L}_2-\mathrm{app},Λ}(N,d)$ on the dimension $d$. The main results are conditions for weak, polynomial, and strong polynomial tractability.