Continuous algorithms in adaptive sampling recovery

Abstract

We study optimal algorithms in adaptive sampling recovery of smooth functions defined on the unit d-cube I d := [0, 1] d . The recovery error is measured in the quasi-norm · q of L q := L q (I d ). For B a subset in L q , we define a sampling recovery algorithm with the free choice of sample points and recovering functions from B as follows. For each f from the quasinormed Besov space B α p,θ , we choose n sample points. This choice defines n sampled values. Based on these sample points and 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 ∈ B α p,θ defines a n-sampling algorithm S B n by functions in B. We suggest a new approach to investigate the optimal adaptive sampling recovery by S B n in the sense of continuous non-linear n-widths which is related to n-term approximation. If Φ = {ϕ k } k∈K is a family of elements in L q , let Σ n (Φ) be the non-linear set of linear combinations of n free terms from Φ, that is Σ n (Φ) := { ϕ = n j=1 a j ϕ kj : k j ∈ K }. Denote by G the set of all families Φ in L q such that the intersection of Φ with any finite dimensional subspace in L q is a finite set, and by C(B α p,θ , L q ) the set of all continuous mappings from B α p,θ into L q . We define the quantityLet 0 < p, q, θ ≤ ∞ and α > d/p. Then we prove the asymptotic orderWe also obtained the asymptotic order of quantities of optimal recovery by S B n in terms of best n-term approximation as well of other non-linear n-widths.Keywords Adaptive sampling recovery · n-sampling algorithm · B-spline quasi-interpolant representation · B-spline · Besov space Mathematics Subject Classifications (2000) 41A46 · 41A05 · 41A25 · 42C40