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  • Book; Book chapter; Dataset


  • Authors: Dinh Dũng (2011)

  • Let be a set of n sample points in the d-cube Id≔[0,1]d, and a family of n functions on Id. We define the linear sampling algorithm Ln(Φ,ξ,⋅) for an approximate recovery of a continuous function f on Id from the sampled values f(x1),…,f(xn), by For the Besov class of mixed smoothness α, to study optimality of Ln(Φ,ξ,⋅) inLq(Id) we use the quantity where the infimum is taken over all sets of n sample points and all families in Lq(Id). We explicitly constructed linear sampling algorithms Ln(Φ,ξ,⋅)on the set of sample points ξ=Gd(m)≔{(2−k1s1,…,2−kdsd)∈Id:k1+⋯+kd≤m}, with Φ a family of linear combinations of mixed B-splines which are mixed tensor products of either integer or half integer translated dilations of the centered B-spline of order r. For various 0

  • Book; Book chapter; Dataset


  • Authors: 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; Dataset


  • Authors: Dinh Dũng (2011)

  • We propose an approach to study optimal methods of adaptive sampling recovery of functions by sets of a finite capacity which is measured by their cardinality or pseudo-dimension. Let W ⊂ L q , 0 < q ≤ ∞ , be a class of functions on Id:=[0,1]dId:=[0,1]d. For B a subset in L q , we define a sampling recovery method with the free choice of sample points and recovering functions from B as follows. For each f ∈ W we choose n sample points. This choice defines n sampled values. Based on these sampled values, we choose a function from B for recovering f. The choice of n sample points and a recovering function from B for each f ∈ W defines a sampling recovery method SBnSnB by functions inB. An efficient sampling recovery method should be adaptive to f. Given a family BB of subsets in Lq , ...

  • Book; Book chapter; Dataset


  • Authors: 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 ...