Uniform discretization of continuous frames

Abstract

Let $H$ be an infinite-dimensional separable Hilbert space and let $(X,d,\mu)$ be a metric measure space satisfying the doubling and upper Alhfors regularity conditions at small scale. We prove that every bounded continuous tight frame $\Psi\colon X\rightarrow H$ can be sampled to obtain a frame for $H$, which is uniformly discrete and nearly tight. That is, for every $0<\epsilon<1$, there exist a sampling sequence $\{x_n\}_{n\in\mathbb{N}}$ in $X$ and $r>0$ such that $\inf_{n\neq m}d(x_n,x_m)\geq r$ and $\{\Psi(x_n)\}_{n\in\mathbb{N}}$ is a frame whose ratio of frame bounds is less than $1+\epsilon$. We apply our main result to show that for every nonzero function $g$ in $L^2(\mathbb{R}^d)$ there exists a uniformly discrete set $\Lambda$ such that the corresponding Gabor system $\{e^{2\pi ibx}g(x-a)\}_{(a,b)\in \Lambda}$ is a nearly tight frame. We also prove that if $\psi\in L^2(\mathbb{R})$ satisfies the Calder\'on admissibility condition, then there exists a uniformly discrete set $\Gamma$ such that wavelet system $\{a^{1/2}\psi(ax-b)\}_{(a,b)\in \Gamma}$ is a nearly tight frame. Analogous discretization results for exponential frames and spectral subspaces of elliptic differential operators are presented as well.