Universal Sampling Discretization

Let $X_N$ be an $N$-dimensional subspace of $L_2$ functions on a probability space $(\Omega , \mu )$ spanned by a uniformly bounded Riesz basis $\Phi _N$. Given an integer $1\le v\le N$ and an exponent $1\le p\le 2$, we obtain universal discretization for the integral norms $L_p(\Omega ,\mu )$ of functions from the collection of all subspaces of $X_N$ spanned by $v$ elements of $\Phi _N$ with the number $m$ of required points satisfying $m\ll v(\log N)^2(\log v)^2$. 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.