On the Power of Standard Information for Tractability for L₂- Approximation in the Randomized Setting

Abstract

We study the approximation of multivariate functions from a separable Hilbert space in the randomized setting with the error measured in the weighted L₂ norm. We consider algorithms that use standard information Λ^std consisting of function values or general linear information Λ^all consisting of arbitrary linear functionals. We investigate the equivalences of various notions of algebraic and exponential tractability in the randomized setting for Λ^std and Λ^all for the normalized or absolute error criterion. For the normalized error criterion, we show that the power of Λ^std is the same as that of Λ^all for all notions of exponential tractability and some notions of algebraic tractability without any condition. For the absolute error criterion, we show that the power of Λ^std is the same as that of Λ^all for all notions of algebraic and exponential tractability without any condition. Specifically, we solve Open Problems 98, 101, 102 and almost solve Open Problem 100 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, Zürich, 2012.