2021/01/12 by Wanting Lu, Heping Wang, Lu, Wanting +1
Computer Science · Mathematics · #41A63 #65C05 #65D15 #65Y20 #FOS: Mathematics #Numerical Analysis (math.NA) #cs.NA #math.NA #msc:41A63 #msc:65C05 #msc:65D15 #msc:65Y20
paper · pdf · doi:10.48550/arxiv.2101.05200
23 pages. arXiv admin note: substantial text overlap with arXiv:2101.03665
arxiv created 2021/01/12 · arxiv updated 2021/01/14
We study multivariate approximation in the average case setting with the error measured in the weighted L2 norm. We consider algorithms that use standard information Λ\rm std consisting of function values or general linear information Λ\rm all consisting of arbitrary continuous linear functionals. We investigate the equivalences of various notions of algebraic and exponential tractability for Λ\rm std and Λ\rm all for the absolute error criterion, and show that the power of Λ\rm std is the same as that of Λ\rm 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.