Improved bounds for the eigenvalues of prolate spheroidal wave functions and discrete prolate spheroidal sequences

TL;DR

This paper provides new non-asymptotic bounds on the eigenvalues of prolate spheroidal wave functions and discrete prolate spheroidal sequences, improving understanding of their clustering behavior for signal processing applications.

Contribution

The work introduces two novel non-asymptotic bounds on DPSS eigenvalues and extends these results to continuous prolate spheroidal wave functions, filling gaps in existing theoretical characterizations.

Findings

Established bounds on the number of eigenvalues between  and 1-

Quantified how close the first  eigenvalues are to 1 and the last  eigenvalues are to 0

Numerical experiments validate the tightness of the bounds

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 $\{0,\ldots,N-1\}$. 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 $\epsilon$ and $1-\epsilon$. Also, we obtain bounds detailing how close the first $\approx 2NW$ eigenvalues are to $1$ and how close the last $\approx N-2NW$ 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 $\epsilon$ and $1-\epsilon$.