  In this paper, we first give two uniform asymptotic approximations of the
eigenfunctions of the weighted finite Fourier transform operator, defined by
${\displaystyle \mathcal F_c^{(\alpha)} f(x)=\int_{-1}^1 e^{icxy}
f(y)\,(1-y^2)^{\alpha}\, dy,\,}$ where $ c >0, \alpha > -1$ are two fixed real
numbers. The first uniform approximation is given in terms of a Bessel
function, whereas the second one is given in terms of a normalized Jacobi
polynomial. These eigenfunctions are called generalized prolate spheroidal wave
functions (GPSWFs). By using the uniform asymptotic approximations of the
GPSWFs, we prove the super-exponential decay rate of the eigenvalues of the
operator $\mathcal F_c^{(\alpha)}$ in the case where $0<\alpha < 3/2.$ Finally,
by computing the trace and an estimate of the norm of the operator
${\displaystyle \mathcal Q_c^{\alpha}=\frac{c}{2\pi} \mathcal F_c^{{\alpha}^*}
\mathcal F_c^{\alpha},}$ we give a lower and an upper bound for the counting
number of the eigenvalues of $Q_c^{\alpha},$ when $c>>1.$
