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 Book; Book chapter; DatasetOptimal adaptive sampling recoveryAuthors: 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; Article; DatasetB-spline quasi-interpolant representations and sampling recovery of functions with mixed smoothnessAuthors: 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