Tractability of approximation in the weighted Korobov space in the worst-case setting -- a complete picture

Adrian Ebert; Friedrich Pillichshammer

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arXiv · 2102.01449

Tractability of approximation in the weighted Korobov space in the worst-case setting -- a complete picture

Adrian Ebert·Friedrich Pillichshammer

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Abstract

In this paper, we study tractability of $L_2$-approximation of one-periodic functions from weighted Korobov spaces in the worst-case setting. The considered weights are of product form. For the algorithms we allow information from the class $\Lambda^{{\rm all}}$ consisting of all continuous linear functionals and from the class $\Lambda^{{\rm std}}$, which only consists of function evaluations. We provide necessary and sufficient conditions on the weights of the function space for quasi-polynomial tractability, uniform weak tractability, weak tractability and $(\sigma,\tau)$-weak tractability. Together with the already known results for strong polynomial and polynomial tractability, our findings provide a complete picture of the weight conditions for all current standard notions of tractability.

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BibTeX RIS

Adrian Ebert, Friedrich Pillichshammer. 2021-02-02. Tractability of approximation in the weighted Korobov space in the worst-case setting -- a complete picture. https://arxiv.org/abs/2102.01449

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