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 Book; Book chapter; DatasetContinuous algorithms in adaptive sampling recoveryAuthors: Dinh Dũng (2012)We study optimal algorithms in adaptive continuous sampling recovery of smooth functions defined on the unit d-cube Id≔[0,1]d. Functions to be recovered are in Besov space . The recovery error is measured in the quasi-norm ‖⋅‖q of . For a set A⊂Lq, we define a sampling algorithm of recovery with the free choice of sample points and recovering functions from A as follows. For each , we choose n sample points which define n sampled values of f. Based on these sample points and sampled values, we choose a function from A for recovering f. The choice of n sample points and a recovering function from A for each defines an n-sampling algorithm . We suggest a new approach to investigate the optimal adaptive sampling recovery by in the sense of continuous non-linear n-widths which... Book; Book chapter; DatasetSampling and Cubature on Sparse Grids Based on a B-spline Quasi-InterpolationAuthors: Dinh Dũng (2012)Let Xn={xj}nj=1Xn={xj}j=1n be a set of n points in the d-cube Id:=[0,1]dId:=[0,1]d, and Φn={φj}nj=1Φn={φj}j=1n a family of n functions on IdId. We consider the approximate recovery of functions f on IdId from the sampled values f(x1),…,f(xn)f(x1),…,f(xn), by the linear sampling algorithm Ln(Xn,Φn,f):=∑nj=1f(xj)φj.Ln(Xn,Φn,f):=∑j=1nf(xj)φj.The error of sampling recovery is measured in the norm of the space Lq(Id)Lq(Id)-norm or the energy quasi-norm of the isotropic Sobolev space Wγq(Id)Wqγ(Id) for 10γ>0. Functions f to be recovered are from the unit ball in Besov-type spaces of an anisotropic smoothness, in particular, spaces Bα,βp,θBp,θα,β of a “hybrid” of mixed smoothness α>0α>0 and isotropic smoothness β∈Rβ∈R, and spaces Bap,θBp,θa of a nonuniform mixed smoothness ...