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<p>We extend classical time-frequency limiting analysis, historically applied to one-dimensional finite signals, to the multidimensional (or multi-indexed) 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, then 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 time-bandwidth product and prescribed accuracy. Concretely, our contributions are twofold: (i) we extend Theorem 1.4 of Israel and Mayeli (Appl. Comput. Harmon. Anal. 70:101620, 2024) to the Cartesian discrete setting for higher-dimensional signals; and (ii) within this framework, we develop a multidimensional generalization of the non-asymptotic eigenvalue distribution analysis for prolate matrices from Karnik et al. (Appl. Comput. Harmon. Anal. 46:624–652, 2019). The advances are summarized in Theorem <InternalRef RefID="FPar1">1.1</InternalRef>. We test our theoretical results through numerical experiments in one- and two-dimensional settings. The empirical results confirm the predicted eigenvalue concentration and illustrate potential applications in fast computation for image analysis, multidimensional spectral estimation, and related signal-processing tasks.</p>

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Eigenvalue distribution analysis of multidimensional prolate matrices

  • 期刊论文

  • 发表日期：

    2026年4月29日

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  • Luis Gomez,
  • Jonathan Jaimangal,
  • Azita Mayeli,
  • Tasfia Proma

摘要

We extend classical time-frequency limiting analysis, historically applied to one-dimensional finite signals, to the multidimensional (or multi-indexed) 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, then 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 time-bandwidth product and prescribed accuracy. Concretely, our contributions are twofold: (i) we extend Theorem 1.4 of Israel and Mayeli (Appl. Comput. Harmon. Anal. 70:101620, 2024) to the Cartesian discrete setting for higher-dimensional signals; and (ii) within this framework, we develop a multidimensional generalization of the non-asymptotic eigenvalue distribution analysis for prolate matrices from Karnik et al. (Appl. Comput. Harmon. Anal. 46:624–652, 2019). The advances are summarized in Theorem 1.1. We test our theoretical results through numerical experiments in one- and two-dimensional settings. The empirical results confirm the predicted eigenvalue concentration and illustrate potential applications in fast computation for image analysis, multidimensional spectral estimation, and related signal-processing tasks.