Uniform discretization of continuous frames

Abstract

Let $H$ be an infinite-dimensional separable Hilbert space and let $(X,d,\hat{I}¼)$ be a metric measure space satisfying the doubling and upper Alhfors regularity conditions at small scale. We prove that every bounded continuous tight frame $\hat{I}¨:X\to H$ can be sampled to obtain a frame for $H$, which is uniformly discrete and nearly tight. That is, for every $0<\hat{I}\mu <1$, there exist a sampling sequence $\{x_n{\}}_{n\in \mathbb{N}}$ in $X$ and $r>0$ such that ${\inf }_{n\ne m}d(x_n,x_m)\ge r$ and $\{\hat{I}¨(x_n){\}}_{n\in \mathbb{N}}$ is a frame whose ratio of frame bounds is less than $1+\hat{I}\mu$. 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 $\hat{I}›$ such that the corresponding Gabor system $\{e^{2\ddot{I}€ibx}g(x-a){\}}_{(a,b)\in \hat{I}›}$ is a nearly tight frame. We also prove that if $\ddot{I}ˆ\in L^2(\mathbb{R})$ satisfies the Calderón admissibility condition, then there exists a uniformly discrete set $\hat{I}“$ such that wavelet system $\{a^{1/2}\ddot{I}ˆ(ax-b){\}}_{(a,b)\in \hat{I}“}$ is a nearly tight frame. Analogous discretization results for exponential frames and spectral subspaces of elliptic differential operators are presented as well.

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