On the power of standard information for tractability for L_2-approximation in the randomized setting
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We study approximation of multivariate functions from a separable Hilbert space in the randomized setting with the error measured in the weighted L_2 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 use the weighted least squares regression algorithm to obtain the upper estimates of the minimal randomized error using Λ^ std. We investigate the equivalences of various notions of algebraic and exponential tractability for Λ^ std and Λ^ all for the normalized or absolute error criterion. We show that in the randomized setting for the normalized or absolute error criterion, the power of Λ^ std is the same as that of Λ^ all for all notions of exponential and algebraic tractability without any condition. Specifically, we solve four Open Problems 98, 100-102 as posed by E.Novak and H.Woźniakowski in the book: Tractability of Multivariate Problems, Volume III: Standard Information for Operators, EMS Tracts in Mathematics, Zürich, 2012.
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