Discrete Prolate Spheroidal Wave Functions: Further Spectral Analysis and Some Related Applications

Abstract

For fixed $W\in \big (0,\frac{1}{2}\big )$ and positive integer $N\ge 1$, the discrete prolate spheroidal wave functions (DPSWFs), denoted by $U_{k,W}^N$, $0\le k\le N-1$ form the set of 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 DPSWFs have a wide range of 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 (Slepian in Bell Syst. Tech. J. 57: 1371–1430,1978) 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 behaviour 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, as well as those functions belonging to periodic Sobolev spaces. Finally, we provide the reader with some numerical examples that illustrate the different results of this work.