The discrete prolate spheroidal sequences (DPSSs) are a set of orthonormal sequences in ${\ell }_2(\mathbb{Z})$ which are strictly bandlimited to a frequency band $[-W,W]$ and maximally concentrated in a time interval $\{0,\dots ,N-1\}$. The timelimited DPSSs (sometimes referred to as the Slepian basis) are an orthonormal set of vectors in ${\mathbb{C}}^N$ whose discrete time Fourier transform (DTFT) is maximally concentrated in a frequency band $[-W,W]$. Due to these properties, DPSSs have a wide variety of signal processing applications. The DPSSs are the eigensequences of a timelimit-then-bandlimit operator and the Slepian basis vectors are the eigenvectors of the so-called prolate matrix. The eigenvalues in both cases are the same, and they exhibit a particular clustering behavior – slightly fewer than $2NW$ eigenvalues are very close to 1, slightly fewer than $N-2NW$ eigenvalues are very close to 0, and very few eigenvalues are not near 1 or 0. This eigenvalue behavior is critical in many of the applications in which DPSSs are used. There are many asymptotic characterizations of the number of eigenvalues not near 0 or 1. In contrast, there are very few non-asymptotic results, and these don't fully characterize the clustering behavior of the DPSS eigenvalues. In this work, we establish two novel non-asymptotic bounds on the number of DPSS eigenvalues between ϵ and $1-ϵ$. Also, we obtain bounds detailing how close the first $\approx 2NW$ eigenvalues are to 1 and how close the last $\approx N-2NW$ eigenvalues are to 0. Furthermore, we extend these results to the eigenvalues of the prolate spheroidal wave functions (PSWFs), which are the continuous-time version of the DPSSs. Finally, we present numerical experiments demonstrating the quality of our non-asymptotic bounds on the number of DPSS eigenvalues between ϵ and $1-ϵ$.

离散长球体序列（DPSS）是一组正交序列 ${\ell }_{\mathrm{2个}}（\mathbb{ž}）$ 严格限制在一个频带内 $[-\mathrm{w\; ^}，\mathrm{w\; ^}]$ 并最大程度地集中在一个时间间隔内 $\{0，\dots ，ñ-\mathrm{1个}\}$。限时的DPSS（有时称为Slepian基）是向量中的正交向量集${\mathbb{C}}^{ñ}$ 其离散时间傅里叶变换（DTFT）最大程度地集中在一个频带中 $[-\mathrm{w\; ^}，\mathrm{w\; ^}]$。由于这些特性，DPSS具有广泛的信号处理应用程序。DPSS是时间限制然后带限制算子的本征序列，而Slepian基向量是所谓的prolate矩阵的本征向量。两种情况下的特征值都是相同的，并且它们表现出特定的聚类行为–略小于$\mathrm{2个}ñ\mathrm{w\; ^}$ 特征值非常接近1，比 $ñ-\mathrm{2个}ñ\mathrm{w\; ^}$特征值非常接近0，很少有特征值不接近1或0。在使用DPSS的许多应用中，此特征值行为至关重要。特征值的数量有许多不接近0或1的渐近特征。相比之下，非渐近结果很少，并且这些特征不能完全表征DPSS特征值的聚类行为。在这项工作中，我们在ϵ和之间的DPSS特征值数目上建立了两个新颖的非渐近界。$\mathrm{1个}-ϵ$。此外，我们获得了边界，详细说明了第一个$\approx \mathrm{2个}ñ\mathrm{w\; ^}$ 特征值是1，最后一个值有多接近 $\approx ñ-\mathrm{2个}ñ\mathrm{w\; ^}$特征值是0。此外，我们将这些结果扩展到扁球面波函数（PSWF）的特征值，PSWF是DPSS的连续时间版本。最后，我们提供了数值实验，证明了ϵ和之间DPSS特征值的数量的非渐近界的质量。$\mathrm{1个}-ϵ$。