  Since the early 1960s, the fields of signal processing, data transmission,
channel equalisation, filter design and others have been technologically
developed and modernised as a result of the research carried out by D. Slepian
and his co-authors H. J Landau and H. O Pollack on the time and band-limited
wave system known as discrete and continuous spheroidal waves systems. Our aim
in this paper is to introduce new discrete wave sequences called discrete
Hankel Prolate spheroidal sequences {\bf DHPSS} and their counterparts in the
frequency domain called discrete Hankel Prolate spheroidal wave functions {\bf
DHPSWF} as radial parts of different solutions of a discrete multidimensional
energy maximization problem similar to the one given by D. Slepian and which
will generalize his classical pioneering work. In the meantime, we will ensure
that our new family is the eigenfunctions set of a finite rank integral
operator defined on $L^2(0,\omega),\,0<\omega<1,$ with an associated kernel
given by $\sum_{k=1}^N\phi^{\alpha}_{n}(r)\phi^{\alpha}_{n}(r'),$ where
$\phi^{\alpha}_{n}(r)=\frac{\sqrt{2r}J_{\alpha}(s_n^{(\alpha)}r)}{|J_{\alpha+1}(s^{(\alpha)}_n)|},0\leq
r\leq 1.$ Here $J_\alpha$ is the Bessel function of the first kind and
$(s_n^{(\alpha)})_n$ are the associated positive zeros. In addition, we will
extend the various classical results proposed concerning the decay rate and
spectral distribution associated with the classical case, then we will finish
our work by an application on the Ingham's universal constant which we will
specify with an upper bound estimate.
