# Codes and Expansions (CodEx) Seminar

## Daniel Freeman (St Louis University)Discretizing $L_p$ norms

Given an $n$-dimensional subspace $X$ of $L_p([0,1])$, we are interested in choosing $m$-sampling points which may be used to discretely approximate the $L_p$ norm on the subspace.   In particular, when can $m$ be chosen on the order of $n$?  For the case $p=2$ it is known that $m$ may always be chosen on the order of $n$ as long as the subspace $X$ satisfies a necessary $L_\infty$ bound.  For other values of $p$ with $1\leq p< \infty$, existing sampling methods allow one to choose $m$ on the order of $n$ times some additional log factors.  However, it was not previously known if these log factors were necessary.  We will show for all $1\leq p < 2$ that there exist classes of subspaces of $L_p([0,1])$ where the number of sampling points $m$ cannot be chosen on the order of $n$.  We will conclude the talk with an application which shows that the problem of discretizing the $L_1$ norm can be applied to phase retrieval on finite dimensional Hilbert spaces.  This is joint work with Dorsa Ghoreishi.