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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 ₁ (F, S) ₂ (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.
Hughes et al. (Tue,) studied this question.