Uniform discretization of continuous frames
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866917332781105152 |
|---|---|
| author | Bownik, Marcin Yu, Pu-Ting |
| author_facet | Bownik, Marcin Yu, Pu-Ting |
| contents | Let $H$ be an infinite-dimensional separable Hilbert space and let $(X,d,μ)$ be a metric measure space satisfying the doubling and upper Alhfors regularity conditions at small scale. We prove that every bounded continuous tight frame $Ψ\colon X\rightarrow H$ can be sampled to obtain a frame for $H$, which is uniformly discrete and nearly tight. That is, for every $0<ε<1$, 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 $\{Ψ(x_n)\}_{n\in\mathbb{N}}$ is a frame whose ratio of frame bounds is less than $1+ε$.
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 $Λ$ such that the corresponding Gabor system $\{e^{2πibx}g(x-a)\}_{(a,b)\in Λ}$ is a nearly tight frame. We also prove that if $ψ\in L^2(\mathbb{R})$ satisfies the Calderón admissibility condition, then there exists a uniformly discrete set $Γ$ such that wavelet system $\{a^{1/2}ψ(ax-b)\}_{(a,b)\in Γ}$ is a nearly tight frame. Analogous discretization results for exponential frames and spectral subspaces of elliptic differential operators are presented as well. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_10423 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Uniform discretization of continuous frames Bownik, Marcin Yu, Pu-Ting Functional Analysis Let $H$ be an infinite-dimensional separable Hilbert space and let $(X,d,μ)$ be a metric measure space satisfying the doubling and upper Alhfors regularity conditions at small scale. We prove that every bounded continuous tight frame $Ψ\colon X\rightarrow H$ can be sampled to obtain a frame for $H$, which is uniformly discrete and nearly tight. That is, for every $0<ε<1$, 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 $\{Ψ(x_n)\}_{n\in\mathbb{N}}$ is a frame whose ratio of frame bounds is less than $1+ε$. 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 $Λ$ such that the corresponding Gabor system $\{e^{2πibx}g(x-a)\}_{(a,b)\in Λ}$ is a nearly tight frame. We also prove that if $ψ\in L^2(\mathbb{R})$ satisfies the Calderón admissibility condition, then there exists a uniformly discrete set $Γ$ such that wavelet system $\{a^{1/2}ψ(ax-b)\}_{(a,b)\in Γ}$ is a nearly tight frame. Analogous discretization results for exponential frames and spectral subspaces of elliptic differential operators are presented as well. |
| title | Uniform discretization of continuous frames |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2603.10423 |