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
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 Λstd consisting of function values or general linear information Λall consisting of arbitrary linear functionals. We investigate the equivalences of various notions of algebraic and exponential tractability in the randomized setting for Λstd and Λall for the normalized or absolute error criterion. 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 some notions of algebraic tractability without any condition. For the absolute error criterion, we show that the power of Λstd is the same as that of Λall 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.
This paper gives a kind of sharp simultaneous approximation error estimation of Birkhoff interpolation [Formula: see text], [Formula: see text] where [Formula: see text] and [Formula: see text] is the Birkhoff interpolation based on [Formula: see text] pairs of numbers [Formula: see text] with its Pólya interpolation matrix to be regular. First, based on the integral remainder formula of Birkhoff interpolation, we refer the computation of [Formula: see text] to the norm of an integral operator. Second, we refer the values of [Formula: see text] and [Formula: see text] to two explicit integral expressions and the value of [Formula: see text] to the computation of the maximum eigenvalue of a Hilbert–Schmidt operator. At the same time, we give the corresponding sharp Wirtinger inequality [Formula: see text] and sharp Picone inequality [Formula: see text].
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