Improved bounds for the eigenvalues of prolate spheroidal wave functions and discrete prolate spheroidal sequences
Abstract
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 $\{0,\ldots,N1\}$. 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 timelimitthenbandlimit operator and the Slepian basis vectors are the eigenvectors of the socalled 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 $N2NW$ 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 nonasymptotic results, and these don't fully characterize the clustering behavior of the DPSS eigenvalues. In this work, we establish two novel nonasymptotic bounds on the number of DPSS eigenvalues between $\epsilon$ and $1\epsilon$. Also, we obtain bounds detailing how close the first $\approx 2NW$ eigenvalues are to $1$ and how close the last $\approx N2NW$ eigenvalues are to $0$. Furthermore, we extend these results to the eigenvalues of the prolate spheroidal wave functions (PSWFs), which are the continuoustime version of the DPSSs. Finally, we present numerical experiments demonstrating the quality of our nonasymptotic bounds on the number of DPSS eigenvalues between $\epsilon$ and $1\epsilon$.
 Publication:

arXiv eprints
 Pub Date:
 May 2020
 arXiv:
 arXiv:2006.00427
 Bibcode:
 2020arXiv200600427K
 Keywords:

 Mathematics  Classical Analysis and ODEs;
 15B05
 EPrint:
 29 pages, 3 figures. V2 includes new results on the eigenvalues of the prolate spheroidal wave functions (PSWFs). The title has been modified to reflect this