Required Number of Points in $L_2$ Marcinkiewicz-Zygmund Inequalities

Abstract: We determine, up to absolute constants, the worst-case number of point evaluations required for a weighted $L_2$ Marcinkiewicz-Zygmund inequality for an $m$-dimensional complex function space. If $0&lt;\varepsilon&lt;1$ is the relative distortion, this number is $$Θ\Big(\min\Big{m<sup>2,\frac{m}{\varepsilon<sup>2}\Big}\Big),$$ and exact discretization has the sharp worst-case value $m<sup>2$. While the upper bounds follow from recent constructions, our contribution is the construction of function spaces that are hard to discretize and yield matching lower bounds. We use a trace-variance inequality for weighted subframes of unit-norm tight frames. One such instance is the complete-graph edge frame, which yields a construction in every dimension. Singer equiangular tight frames improve the constant when $m-1$ is a prime power, while maximal equiangular tight frames give the strongest bound possible using our method whenever they exist. We also derive consequences for the conditioning of weighted least-squares systems and for standard condition-number-based iteration estimates when these systems are solved by LSQR.