Eigenvalue distribution analysis of multidimensional prolate matrices
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866911054726955008 |
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| author | Gomez, Luis Jaimangal, Jonathan Mayeli, Azita Proma, Tasfia |
| author_facet | Gomez, Luis Jaimangal, Jonathan Mayeli, Azita Proma, Tasfia |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_10412 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Eigenvalue distribution analysis of multidimensional prolate matrices Gomez, Luis Jaimangal, Jonathan Mayeli, Azita Proma, Tasfia Classical Analysis and ODEs 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. |
| title | Eigenvalue distribution analysis of multidimensional prolate matrices |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2507.10412 |