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

• Published in: Applied and computational harmonic analysis Vol. 70; p. 101620
• Main Authors: Israel, Arie, Mayeli, Azita
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.05.2024
• Subjects: Prolate spheroidal wave functions; Spatio-spectral limiting operators; Spectral analysis; Wave packets; Prolate spheroidal wave functions; Spectral analysis; Spatio-spectral limiting operators; Wave packets
• ISSN: 1063-5203; 1096-603X
• DOI: 10.1016/j.acha.2023.101620

Prolate spheroidal wave functions are an orthogonal family of bandlimited functions on R that have the highest concentration within a specific time interval. They are also identified as the eigenfunctions of a time-frequency limiting operator (TFLO), and the associated eigenvalues belong to the interval [0,1]. Previous work has studied the asymptotic distribution and clustering behavior of the TFLO eigenvalues. In this paper, we extend these results to multiple dimensions. We prove estimates on the eigenvalues of a spatio-spectral limiting operator (SSLO) on L2(Rd), which is an alternating product of projection operators associated to given spatial and frequency domains in Rd. If one of the domains is a hypercube, and the other domain is convex body satisfying a symmetry condition, we derive quantitative bounds on the distribution of the SSLO eigenvalues in the interval [0,1]. To prove our results, we design an orthonormal system of wave packets in L2(Rd) that are highly concentrated in the spatial and frequency domains. We show that these wave packets are “approximate eigenfunctions” of a spatio-spectral limiting operator. To construct the wave packets, we use a variant of the Coifman-Meyer local sine basis for L2[0,1], and we lift the basis to higher dimensions using a tensor product.