  Recently, with the progress of science and the characteristic properties that
distinguish the Slepian system called Prolate spheroidal wave functions from
the others orthonormal systems, it became clear its important contributions in
several areas such as mathematical statistics, signal processing, numerical
analysis etc...The main issue of this work is to establish the convergence
quality of the truncated error of a function $f$ from the Hankel Sobolev space
$\mathcal{H}_{\alpha}^r$, $r > 0$ and $\alpha\geq -1/2$, by a generalized
prolate spheroidal wave functions basis in $L^2$ norm. In the meantime, we will
create two uniform approximations of the previous system by two special
functions, the first is a Bessel function type and the second is a modified
Jacobi polynomial. Furthermore, using the fact that the generalized PSWFs are
the eigenfunctions of a Sturm Liouville operator, we will improve a new bound
of the associated eigenvalues. The generalized prolate basis we consider here
is the circular prolate spheroidal wave function (CPSWFs), called also Hankel
prolate, introduced by D. Slepian in 1964, denoted by $\vp$, $c > 0$. The
Hankel prolate covers the classical PSWFs $(\alpha=\pm1/2)$ for which
corresponding results have been established previously by many authors.
