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 l 2 ( Z ) which are strictly bandlimited to a frequency band [ − W , W ] and maximally concentrated in a time interval { 0 , … , N − 1 } . The timelimited DPSSs (sometimes referred to as the Slepian basis) are an orthonormal set of vectors in 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 2 N W eigenvalues are very close to 1, slightly fewer than N − 2 N W 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 ≈ 2 N W eigenvalues are to 1 and how close the last ≈ N − 2 N W 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 .