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Frame redundancy and Beurling density

Marcin Bownik, Jordy Timo van Velthoven

TL;DR

This work establishes a quantitative link between frame redundancy and Beurling density for a broad class of reproducing kernel Hilbert spaces on metric measure spaces. By proving that the frame measure function equals the reciprocal Beurling density, and showing that frames with density above the critical threshold contain subframes with density arbitrarily close to one, the authors achieve near-optimal sampling results across multiple settings. The methods combine a selector form of Weaver's conjecture with discretization and Beurling-density techniques, yielding near-critical-density frames for exponential, Gabor, coherent-state, and elliptic-operator spectral spaces, as well as weighted spaces of entire functions. These results generalize and optimize previous density conditions, providing a unified framework for optimal sampling and redundancy analysis in infinite-frame contexts with wide applicability in harmonic analysis and signal processing.

Abstract

We show that the frame measure function of a frame in certain reproducing kernel Hilbert spaces on metric measure spaces is given by the reciprocal of the Beurling density of its index set. In addition, we show that each such frame with Beurling density greater than one contains a subframe with Beurling density arbitrary close to one. This confirms that the concept of frame measure function as introduced by Balan and Landau is a meaningful quantitative definition for the redundancy of a large class of infinite frames. In addition, it shows that the necessary density conditions for sampling in reproducing kernel Hilbert spaces obtained by Führ, Gröchenig, Haimi, Klotz and Romero are optimal. As an application, we also settle the open questions of the existence of frames near the critical density for exponential frames on unbounded sets and for nonlocalized Gabor frames. The techniques used in this paper combine a selector form of Weaver's conjecture and various methods for quantifying the overcompleteness of frames.

Frame redundancy and Beurling density

TL;DR

This work establishes a quantitative link between frame redundancy and Beurling density for a broad class of reproducing kernel Hilbert spaces on metric measure spaces. By proving that the frame measure function equals the reciprocal Beurling density, and showing that frames with density above the critical threshold contain subframes with density arbitrarily close to one, the authors achieve near-optimal sampling results across multiple settings. The methods combine a selector form of Weaver's conjecture with discretization and Beurling-density techniques, yielding near-critical-density frames for exponential, Gabor, coherent-state, and elliptic-operator spectral spaces, as well as weighted spaces of entire functions. These results generalize and optimize previous density conditions, providing a unified framework for optimal sampling and redundancy analysis in infinite-frame contexts with wide applicability in harmonic analysis and signal processing.

Abstract

We show that the frame measure function of a frame in certain reproducing kernel Hilbert spaces on metric measure spaces is given by the reciprocal of the Beurling density of its index set. In addition, we show that each such frame with Beurling density greater than one contains a subframe with Beurling density arbitrary close to one. This confirms that the concept of frame measure function as introduced by Balan and Landau is a meaningful quantitative definition for the redundancy of a large class of infinite frames. In addition, it shows that the necessary density conditions for sampling in reproducing kernel Hilbert spaces obtained by Führ, Gröchenig, Haimi, Klotz and Romero are optimal. As an application, we also settle the open questions of the existence of frames near the critical density for exponential frames on unbounded sets and for nonlocalized Gabor frames. The techniques used in this paper combine a selector form of Weaver's conjecture and various methods for quantifying the overcompleteness of frames.

Paper Structure

This paper contains 19 sections, 32 theorems, 243 equations.

Table of Contents

  1. Introduction
  2. Necessary density conditions
  3. Frames near the critical density
  4. Frame redundancy
  5. Methods
  6. Organization
  7. Frames in Hilbert spaces
  8. Reproducing kernels on metric measure spaces
  9. Assumptions
  10. Sets and Beurling densities
  11. Frame measure and Beurling density
  12. Frames near the critical density
  13. Examples
  14. Exponential frames
  15. Gabor systems
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $\mathcal{H} \subseteq L^2 (\mathbb{R}^n)$ be a reproducing kernel Hilbert space with reproducing kernels $\{k_x \}_{x \in \mathbb{R}^n}$. Suppose that $\{k_x \}_{x \in \mathbb{R}^n}$ satisfies the diagonal condition eq:dc_intro, the weak localization eq:wl_intro and the homogeneous approximatio In particular, if $\{k_{\lambda} \}_{\lambda \in \Lambda}$ is a Riesz basis for $\mathcal{H}$, then

Theorems & Definitions (55)

  • Theorem 1.1: fuehr2017density
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 2.1: bownik2024selector
  • Theorem 2.2: bownik2024selector
  • Lemma 3.1: fuehr2017density
  • Lemma 3.2
  • proof
  • ...and 45 more