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

Peter Kritzer; Helene Laimer; Friedrich Pillichshammer

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arXiv · 1701.02910

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

Peter Kritzer·Helene Laimer·Friedrich Pillichshammer

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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 $e^{\mathbb{L}_2-\mathrm{app},\Lambda}(N,d)$ of all algorithms that use $N$ information evaluations from the class $\Lambda$ in the $d$-dimensional case. The two classes $\Lambda$ considered in this paper are the class $\Lambda^{{\rm all}}$ consisting of all linear functionals and the class $\Lambda^{{\rm std}}$ consisting only of function evaluations. The focus lies on the dependence of $e^{\mathbb{L}_2-\mathrm{app},\Lambda}(N,d)$ on the dimension $d$. The main results are conditions for weak, polynomial, and strong polynomial tractability.

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Peter Kritzer, Helene Laimer, Friedrich Pillichshammer. 2017-01-11. Tractability of $\mathbb{L}_2$-approximation in hybrid function spaces. https://arxiv.org/abs/1701.02910

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