On the power of standard information for tractability for L_2-approximation in the average case setting
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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 Λ^ std consisting of function values or general linear information Λ^ all consisting of arbitrary continuous linear functionals. We investigate the equivalences of various notions of algebraic and exponential tractability for Λ^ std and Λ^ all for the absolute error criterion, and 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 116-118 and almost solve Open Problem 115 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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