Universal Sampling Discretization

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