On the power of standard information for [formula omitted]-approximation in the average case setting

Abstract

This paper studies the equivalence of tractability for classes of Λstd of function evaluations and Λall of all continuous linear functionals for L2-approximation in the average-case setting. For the normalized error criterion, we show that the power of Λstd is the same as that of Λall for all notions of exponential tractability and algebraic tractability, in the sense that there exist algorithms using function values at some deterministic points which enjoy the same tractability properties as those using optimally chosen linear functionals. Moreover, for the absolute error criterion, we also show that the power of Λstd is the same as that of Λall for all notions of exponential tractability and algebraic tractability under certain assumptions on the trace of the covariance operators. In particular, we give a partial answer to three problems raised by E. Novak and H. Woźniakowski in the book “Tractability of multivariate problems” (Open problems 116-118, EMS Tracts in Mathematics, vol.18, Zürich, 2012), proving that the same results hold under weaker conditions. Finally, we apply our results to the tensor product L2-approximation.