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

TL;DR

This paper demonstrates that standard information is as powerful as arbitrary linear information for multivariate $L_2$-approximation in the average case, establishing equivalences in various notions of tractability and resolving open problems in the field.

Contribution

It proves the equivalence of standard and linear information in all tractability notions for multivariate $L_2$-approximation, solving several open problems from prior research.

Findings

Standard information matches the power of all linear information for tractability.

Equivalence holds for both algebraic and exponential tractability.

The results resolve key open problems in the theory of multivariate approximation.

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.