On the Eigenvalue Distribution of Spatio-Spectral Limiting Operators in Higher Dimensions, II

Hughes, Kevin; Israel, Arie; Mayeli, Azita

Abstract

[Abstract is missing mathematical symbols.]

Let F, S be bounded measurable sets in . Let be the orthogonal projection on the subspace of functions with compact support on F, and let be the orthogonal projection on the subspace of functions with Fourier transforms having compact support on S. In this paper, we derive distributional estimates on the eigenvalue sequence of the spatio-spectral limiting operator

. The significance of such estimates lies in their diverse applications in medical imaging, signal processing, geophysics and astronomy. For suitable domains F and S, we prove that

where
represents the Lebesgue measure of the domain

, and the error term satisfies the following bound:

where
and denote the -dimensional Hausdorff measures of the boundaries of F and S, while are geometric constants related to an Ahlfors regularity condition on the domain boundaries. When F and S are Euclidean balls, we expect this estimate to be sharp up to logarithmic factors. This improves on recent work of Marceca-Romero-Speckbacher (Arch. Rational Mech. Anal., 2024) which showed that for any . Our proof is based on the decomposition techniques developed by Marceca-Romero-Speckbacher. The novelty of our approach lies in the use of a two-stage dyadic decomposition with respect to both the spatial and frequency domains, and the application of results in the authors’ prior work on the eigenvalues of spatio-spectral limiting operators associated to cubical domains.

Citation

Hughes, K., Israel, A., & Mayeli, A. (2025). On the Eigenvalue Distribution of Spatio-Spectral Limiting Operators in Higher Dimensions, II. Journal of Fourier Analysis and Applications, 31(4), Article 51. https://doi.org/10.1007/s00041-025-10171-w

• Journal Article Type: Article
• Acceptance Date: Feb 16, 2025
• Online Publication Date: Jul 8, 2025
• Publication Date: 2025-08
• Deposit Date: Aug 5, 2025
• Journal: Journal of Fourier Analysis and Applications
• Print ISSN: 1069-5869
• Electronic ISSN: 1531-5851
• Publisher: Springer
• Peer Reviewed: Peer Reviewed
• Volume: 31
• Issue: 4
• Article Number: 51
• DOI: https://doi.org/10.1007/s00041-025-10171-w