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 $\Lambda^{\rm std}$ consisting of function values or general linear information $\Lambda^{\rm all}$ consisting of arbitrary continuous linear functionals. We investigate the equivalences of various notions of algebraic and exponential tractability for $\Lambda^{\rm std}$ and $\Lambda^{\rm all}$ for the absolute error criterion, and show that the power of $\Lambda^{\rm std}$ is the same as that of $\Lambda^{\rm all}$ 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.Wo\'zniakowski in the book: Tractability of Multivariate Problems, Volume III: Standard Information for Operators, EMS Tracts in Mathematics, Z\"urich, 2012.