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

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 ${\Lambda }^{\mathrm{std}}$ consisting of function values or general linear information ${\Lambda }^{\mathrm{all}}$ consisting of arbitrary continuous linear functionals. We investigate the equivalences of various notions of algebraic and exponential tractability for ${\Lambda }^{\mathrm{std}}$ and ${\Lambda }^{\mathrm{all}}$ for the absolute error criterion, and show that the power of ${\Lambda }^{\mathrm{std}}$ is the same as that of ${\Lambda }^{\mathrm{all}}$ 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.

23 pages. arXiv admin note: substantial text overlap with arXiv:2101.03665