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.