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arXiv · 2101.05200

On the power of standard information for tractability for $L_2$-approximation in the average case setting

Wanting Lu·Heping Wang
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Abstract

We study multivariate approximation in the average case setting with the error measured in the weighted $L_2$ norm. We consider algorithms that use standard information $Λ^{\rm std}$ consisting of function values or general linear information $Λ^{\rm all}$ consisting of arbitrary continuous linear functionals. We investigate the equivalences of various notions of algebraic and exponential tractability for $Λ^{\rm std}$ and $Λ^{\rm all}$ for the absolute error criterion, and show that the power of $Λ^{\rm std}$ is the same as that of $Λ^{\rm all}$ for all notions of algebraic and exponential tractability without any condition. Specifically, we solve Open Problems 116-118 and almost solve Open Problem 115 as posed by E.Novak and H.Woźniakowski in the book: Tractability of Multivariate Problems, Volume III: Standard Information for Operators, EMS Tracts in Mathematics, Zürich, 2012.

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Wanting Lu, Heping Wang. 2021-01-12. On the power of standard information for tractability for $L_2$-approximation in the average case setting. https://arxiv.org/abs/2101.05200

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