  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.
