Continuous algorithms in adaptive sampling recovery

• Main Author: Dinh Dũng
• Format: Book Book chapter Dataset
• Published: Journal of Approximation Theory 2016
• Subjects: Adaptive sampling recovery; Continuous n-sampling algorithm; B-spline quasi-interpolant representation; Besov space
• Online Access: http://repository.vnu.edu.vn/handle/VNU_123/10983
• Institution: Vietnam National University, Hanoi

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 . The recovery error is measured in the quasi-norm ‖⋅‖q of . 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 , we choose n sample points which define n sampled values of f. Based on these sample points and sampled values, we choose a function from A for recovering f. The choice of n sample points and a recovering function from A for each defines an n-sampling algorithm . We suggest a new approach to investigate the optimal adaptive sampling recovery by 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 the set of all continuous mappings from into Lq. We define the quantity For 0<p,q,θ≤∞ and α>d/p, we prove the asymptotic order