Universal sampling discretization

Abstract

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 $1\le q\le 2$, we obtain universal discretization for integral norms $L_q(\Omega ,\mu )$ 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.

References in corpus (1)

• Sampling discretization of integral norms