Optimal adaptive sampling recovery

Abstract

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 , we consider optimal methods of adaptive sampling recovery of functions in W by B from BB in terms of the quantity Rn(W,B)q:= infB∈Bsupf∈WinfSBn∥f−SBn(f)∥q.Rn(W,B)q:= infB∈Bsupf∈WinfSnB∥f−SnB(f)∥q. Denote Rn(W,B)qRn(W,B)q by e n (W) q if BB is the family of all subsets B of L q such that the cardinality ofB does not exceed 2 n , and by r n (W) q if BB is the family of all subsets B in L q 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,θUp,θα (defined as the unit ball of the Besov space Bαp,θBp,θα), there is the following asymptotic order en(Uαp,θ)q ≍ rn(Uαp,θ)q ≍ n−α/d.en(Up,θα)q ≍ rn(Up,θα)q ≍ n−α/d. To construct asymptotically optimal adaptive sampling recovery methods for en(Uαp,θ)qen(Up,θα)q and rn(Uαp,θ)qrn(Up,θα)q we use a quasi-interpolant wavelet representation of functions in Besov spaces associated with some equivalent discrete quasi-norm.