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 I^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 S^B_n by functions in B. An efficient sampling recovery method should be adaptive to f. Given a family B of subsets in L_q, we consider optimal methods of adaptive sampling recovery of functions in W by B from B in terms of the quantity Denote R_n(W,B)_q by e_n(W)_q if B is the family of all subsets B of L_q such that the cardinality of B does not exceed 2ⁿ, and by r_n(W)_q if B 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 U^α_p,θ (defined as the unit ball of the Besov space B^α_p,θ), there is the following asymptotic order To construct asymptotically optimal adaptive sampling recovery methods for e_n(U^α_p,θ)_q and r_n(U^α_p,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.

Dung D.

26.pdf    Gửi cho bạn bè

Từ khóa :