Discrete Prolate Spheroidal Wave Functions: Further spectral analysis and some related applications

TL;DR

This paper advances the spectral analysis of discrete prolate spheroidal wave functions (DPSWFs), providing new decay rate bounds, spectral properties, and demonstrating their effectiveness in approximating band-limited and Sobolev space functions.

Contribution

The work introduces new non-asymptotic decay bounds for DPSWF eigenvalues and extends classical spectral results from PSWFs to DPSWFs, enhancing their theoretical understanding and practical applications.

Findings

Eigenvalues are bounded by classical PSWF eigenvalues up to a small constant.

DPSWFs effectively approximate band-limited functions.

Numerical examples illustrate the theoretical spectral bounds.

Abstract

For fixed $W\in \big(0,\frac{1}{2}\big)$ and positive integer $N\geq 1,$ the discrete prolate spheroidal wave functions (DPSWFs), denoted by $U_{k,W}^N,$ $0\leq k\leq N-1$ form the set of the eigenfunctions of the positive and finite rank integral operator $\widetilde Q_{N,W},$ defined on $L^2(-1/2,1/2),$ with kernel $K_N(x,y)=\frac{\sin(N\pi(x-y))}{\sin(\pi(x-y))}\, \mathbf 1_{[-W,W]}(y).$ It is well known that the DPSWF's have a wide range of classical as well as recent signal processing applications. These applications rely heavily on the properties of the DPSWFs as well as the behaviour of their eigenvalues $\widetilde \lambda_{k,N}(W).$ In his pioneer work \cite{Slepian}, D. Slepian has given the properties of the DPSWFs, their asymptotic approximations as well as the asymptotic behaviour and asymptotic decay rate of these eigenvalues. In this work, we give further properties as well as new non-asymptotic decay rates of the spectrum of the operator $\widetilde Q_{N,W}.$ In particular, we show that each eigenvalue $\widetilde \lambda_{k,N}(W)$ is up to a small constant bounded above by the corresponding eigenvalue, associated with the classical prolate spheroidal wave functions (PSWFs). Then, based on the well established results concerning the distribution and the decay rates of the eigenvalues associated with the PSWFs, we extend these results to the eigenvalues $\widetilde \lambda_{k,N}(W)$. Also, we show that the DPSWFs can be used for the approximation of classical band-limited functions and they are well adapted for the approximation of functions from periodic Sobolev spaces. Finally, we provide the reader with some numerical examples that illustrate the different results of this work.