On the power of standard information for tractability for L2-approximation in the average case setting

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(std) consisting of function values or general linear information Lambda(all) consisting of arbitrary continuous linear functionals. We investigate the equivalences of various notions of algebraic and exponential tractability for Lambda(std) and Lambda(all) for the absolute error criterion, and show that the power of Lambda(std) is the same as that of Lambda(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 Novak and Wozniakowski (2012) [25]. (C) 2021 Elsevier Inc. All rights reserved.

Automated summary

We study multivariate approximation in the average case setting with the error measured in the weighted L-2 norm. We investigate the equivalences of various notions of algebraic and exponential tractability for standard information and general linear information for the absolute error criterion, and show that the power of standard information is the same as that of general linear information for all notions of algebraic and exponential tractability without any condition.