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 ${{\mathbb I}}^d:= [0,1]^d$. 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 $S_n^B$ by functions in B. An efficient sampling recovery method should be adaptive to f. Given a family ${\mathcal B}$ of subsets in L _q , we consider optimal methods of adaptive sampling recovery of functions in W by B from ${\mathcal B}$ in terms of the quantity

$$ R_n(W, {\mathcal B})_q := \ \inf_{B \in {\mathcal B}}\, \sup_{f \in W} \, \inf_{S_n^B} \, \|f - S_n^B(f{\kern1pt})\|_q. $$

Denote $R_n(W, {\mathcal B})_q$ by e _n (W) _q if ${\mathcal B}$ 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 ${\mathcal B}$ 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 $U^\alpha_{p,\theta}$ (defined as the unit ball of the Besov space $B^\alpha_{p,\theta}$), there is the following asymptotic order

$$ e_n\big(U^\alpha_{p,\theta}\big)_q \ \asymp \ r_n\big(U^\alpha_{p,\theta}\big)_q \ \asymp \ n^{- \alpha / d} . $$

To construct asymptotically optimal adaptive sampling recovery methods for $e_n(U^\alpha_{p,\theta})_q$ and $r_n(U^\alpha_{p,\theta})_q$ we use a quasi-interpolant wavelet representation of functions in Besov spaces associated with some equivalent discrete quasi-norm.