Eigenvalue distribution analysis of multidimensional prolate matrices

We extend classical time-frequency limiting analysis, historically applied to one-dimensional finite signals, to the multidimensional discrete setting. This extension is relevant for images, videos, and other multidimensional signals, as it enables a rigorous study of joint time-frequency localization in higher dimensions. To achieve this, we define multidimensional time-limiting and frequency-limiting matrices tailored to signals on a Cartesian grid and construct a multi-indexed prolate matrix. We prove that the spectrum of this matrix exhibits an eigenvalue concentration phenomenon: the bulk of eigenvalues cluster near 1 or 0 with a narrow transition band separating these regions. Moreover, we derive quantitative bounds on the width of the transition band in terms of the time-bandwidth product and prescribed accuracy. Concretely, our contributions are twofold: (i) we extend existing one-dimensional results to higher-dimensional Cartesian discrete signals; and (ii) we develop a multidimensional non-asymptotic eigenvalue-distribution analysis for prolate matrices. The advances are summarized in Theorem 1.1. Numerical experiments in one- and two-dimensional settings confirm the predicted eigenvalue concentration and illustrate potential applications in fast computation for image analysis, multidimensional spectral estimation, and related signal-processing tasks.