On the eigenvalue distribution of spatio-spectral limiting operators in higher dimensions, II

Let F, S be bounded measurable sets in Rᵈ. Let PF : L²(Rᵈ) → L²(Rᵈ) be the orthogonal projection on the subspace of functions with compact support on F, and let BS : L²(Rᵈ) → L²(Rᵈ) be the orthogonal projection on the subspace of functions with Fourier transforms having compact support on S. In this paper, we derive improved distributional estimates on the eigenvalue sequence 1 ≥ λ₁(F,S) ≥ λ₂(F,S) ≥ ⋯ > 0 of the spatio-spectral limiting operator BS PF BS : L²(Rᵈ) → L²(Rᵈ). The significance of such estimates lies in their diverse applications in medical imaging, signal processing, geophysics and astronomy. Our proof is based on the decomposition techniques developed in {MaRoSp23}. The novelty of our approach is in the use of a two-stage dyadic decomposition with respect to both the spatial and frequency domains, and the application of the results in {ArieAzita23} on the eigenvalues of spatio-spectral limiting operators associated to cubical domains.