Universal sampling discretization

07/23/2021

∙

by   Feng Dai, et al.

∙

0

∙

share

Let X_N be an N-dimensional subspace of L_2 functions on a probability space (Ω, μ) spanned by a uniformly bounded Riesz basis Φ_N. Given an integer 1≤ v≤ N and an exponent 1≤ q≤ 2, we obtain universal discretization for integral norms L_q(Ω,μ) of functions from the collection of all subspaces of X_N spanned by v elements of Φ_N with the number m of required points satisfying m≪ v(log N)^2(log v)^2. This last bound on m is much better than previously known bounds which are quadratic in v. Our proof uses a conditional theorem on universal sampling discretization, and an inequality of entropy numbers in terms of greedy approximation with respect to dictionaries.