Continuous algorithms in adaptive sampling recovery

Abstract

We study optimal algorithms in adaptive continuous sampling recovery of smooth functions defined on the unit d-cube Id≔[0,1]d. Functions to be recovered are in Besov space Bp,θα. The recovery error is measured in the quasi-norm ‖⋅‖q of Lq≔Lq(Id),0<q≤∞. For a set A⊂Lq, we define a sampling algorithm of recovery with the free choice of sample points and recovering functions from A as follows. For each f∈Bp,θα, we choose n sample points which define n sampled values of f. Based on these sample points and sampled values, we choose a function SnA(f) from A for recovering f. The choice of n sample points and a recovering function from A for each f∈Bp,θα defines an n-sampling algorithm SnA. We suggest a new approach to investigate the optimal adaptive sampling recovery by SnA in the sense of continuous non-linear n-widths which is related to n-term approximation. If Φ={φk}k∈K is a family of functions in Lq, let Σn(Φ) be the non-linear set of linear combinations of n free terms from Φ. Denote by G the set of all families Φ such that the intersection of Φ with any finite dimensional subspace in Lq is a finite set, and by C(Bp,θα,Lq) the set of all continuous mappings from Bp,θα into Lq. We define the quantity νn(Bp,θα,Lq)≔infΦ∈GinfSnA∈C(Bp,θα,Lq):A=Σn(Φ)sup‖f‖Bp,θα≤1‖f−SnA(f)‖q. For 0<p,q,θ≤∞ and α>d/p, we prove the asymptotic order νn(Bp,θα,Lq)≍n−α/d.