Uniform discretization of continuous frames

TL;DR

This paper demonstrates that continuous frames in Hilbert spaces can be sampled to produce nearly tight, uniformly discrete frames, with applications to Gabor and wavelet systems, advancing discretization techniques in harmonic analysis.

Contribution

It introduces a method to discretize continuous frames into nearly tight, uniformly discrete frames in Hilbert spaces, with broad applications to Gabor and wavelet systems.

Findings

Existence of uniformly discrete sampling sets for continuous frames.

Construction of nearly tight frames from continuous frames.

Applications to Gabor and wavelet systems.

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.