论文标题
改进的岩体球波函数和离散的球形序列的特征值的边界
Improved bounds for the eigenvalues of prolate spheroidal wave functions and discrete prolate spheroidal sequences
论文作者
论文摘要
离散的pr酸球体序列（DPSSS）是$ \ ell_2（\ Mathbb {Z}）$中的一组正顺序序列，这些序列严格限制为频段$ [ - w，w，w] $，并最大程度地集中在时间间隔$ \ \ \ ldots，\ ldots n n n n n-1，n-1，n-1，n-1，n-1} $中。时间序列的DPSS（有时称为SLEPIAN基础）是$ \ Mathbb {C}^n $中的正顺序矢量集，其离散时间傅立叶变换（DTFT）最大地集中在频段$ [-W，W，W，w] $中。由于这些属性，DPSS具有多种信号处理应用。 dpss是时间元素的特征 - 然后是伴随限速操作员，而slepian基矢量是所谓的pr酸矩阵的特征向量。在这两种情况下，特征值都是相同的，并且表现出特殊的聚类行为 - 略低于$ 2nw $ eigenvalues非常接近$ 1 $，略少于$ n-2nw $ eigenvalues非常接近$ 0 $，而且很少有特征值接近$ 1 $或$ 0 $。这种特征值行为在使用DPSS的许多应用中至关重要。特征值的数量不接近$ 0 $或$ 1 $。相比之下，非反应结果很少，并且这些结果并未完全表征DPSS特征值的聚类行为。在这项工作中，我们在$ε$和$1-ε$之间的dpss特征值数量上建立了两个新颖的非反应界限。此外，我们获得了详细信息，详细介绍了第一个$ \ 2nw $ eigenvalues到$ 1 $的近距离，以及最后的$ \ \ \ n-2nw $ eigenvalues到$ 0 $。此外，我们将这些结果扩展到了pr素球波函数（PSWF）的特征值，这是DPSSS的连续时间版本。最后，我们介绍了数值实验，以证明我们非扰动界限的质量在$ε$和$1-ε$之间的DPSS特征值数量上。
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 $\{0,\ldots,N-1\}$. 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 $1-ε$.
