标题： 多维椭圆矩阵的特征值分布分析 显示英文标题

标题： Eigenvalue distribution analysis of multidimensional prolate matrices

摘要： 我们扩展了经典的时频限制分析，该分析历史上用于一维有限信号，以适用于多维离散设置。 这种扩展对于图像、视频和其他多维信号是相关的，因为它使得在更高维度中对联合时频局部化的严格研究成为可能。 为了实现这一点，我们定义了针对笛卡尔网格上信号的多维时域限制和频域限制矩阵，并构建了一个多索引的勒让德矩阵。 我们证明了该矩阵的谱表现出特征值集中现象：大部分特征值集中在1或0附近，且有一个狭窄的过渡带将这些区域分隔开。 此外，我们推导了过渡带宽度的定量界限，以时间带宽乘积和预定精度为基准。 具体而言，我们的贡献有两个方面：(i) 我们将现有的二维结果扩展到更高维度的笛卡尔离散信号；以及 (ii) 我们为勒让德矩阵开发了多维非渐近特征值分布分析。 这些进展在定理1.1中进行了总结。 一维和二维设置中的数值实验验证了预测的特征值集中现象，并展示了在图像分析快速计算、多维谱估计及相关信号处理任务中的潜在应用。 显示英文摘要

摘要： 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.