Optimal adaptive sampling recovery
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 ⊂ Lq, 0 < q ≤ ∞, be a class of functions on Id. For B a subset in Lq, 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 SBn by functions in B. An efficient sampling recovery method should be adaptive to f. Given a family B of subsets in Lq, we consider optimal methods of adaptive sampling recovery of functions in W by B from B in terms of the quantity Denote Rn(W,B)q by en(W)q if B is the family of all subsets B of Lq such that the cardinality of B does not exceed 2n, and by rn(W)q if B is the family of all subsets B in Lq of pseudo-dimension at most n. Let 0 < p,q, θ ≤ ∞ and α satisfy one of the following conditions: (i) α > d/p; (ii) α = d/p, θ ≤ min (1,q), p,q < ∞. Then for the d-variable Besov class Uαp,θ (defined as the unit ball of the Besov space Bαp,θ), there is the following asymptotic order To construct asymptotically optimal adaptive sampling recovery methods for en(Uαp,θ)q and rn(Uαp,θ)q we use a quasi-interpolant wavelet representation of functions in Besov spaces associated with some equivalent discrete quasi-norm. © 2009 Springer Science + Business Media, LLC.