Frame redundancy and Beurling density

2025/09/15 by Marcin Bownik, Bownik, Marcin, Jordy Timo van Velthoven +1 · 2 citations
Engineering · #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA) #Structural Load-Bearing Analysis

paper · pdf · doi:10.48550/arxiv.2509.11887

openalex publication_date 2025/09/15 · openalex created_date 2025/10/12 · openalex updated_date 2026/07/28

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.

Citations

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Cited by

• Exponential frames and Riesz sequences at the critical density
• Universal frame set for rational functions

Related

• A quantitative notion of redundancy for finite frames
• Redundancy for localized and Gabor frames
• Frames, their relatives and reproducing kernel Hilbert spaces
• A quantitative notion of redundancy for infinite frames
• Measure Functions for Frames