Discretizing $L_p$ norms and frame theory

Given an $N$-dimensional subspace $X$ of $L_p([0,1])$, we consider the problem of choosing $M$-sampling points which may be used to discretely approximate the $L_p$ norm on the subspace. We are particularly interested in knowing when the number of sampling points $M$ can be chosen on the order of the dimension $N$. 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 natural $L_\infty$ bound, and for the case $p=\infty$ there are examples where $M$ may not be chosen on the order of $N$. We show for all $1\leq p<2$ that there exist classes of subspaces of $L_p([0,1])$ which satisfy the $L_\infty$ bound, but where the number of sampling points $M$ cannot be chosen on the order of $N$. We show as well that the problem of discretizing the $L_p$ norm of subspaces is directly connected with frame theory. In particular, we prove that discretizing a continuous frame to obtain a discrete frame which does stable phase retrieval requires discretizing both the $L_2$ norm and the $L_1$ norm on the range of the analysis operator of the continuous frame.