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For a fixed reals $c>0$, $a>0$ and $\alpha>-\frac{1}{2}$, the circularprolate spheroidal wave functions (CPSWFs) or 2d-Slepian functions as someauthors call it, are the eigenfunctions of the finite Hankel transformoperator, denoted by $\mathcal{H}_c^{\alpha}$, which is the integral operatordefined on $L^2(0,1)$ with kernel$H_c^{\alpha}(x,y)=\sqrt{cxy}J_{\alpha}(cxy)$. Also, they are theeigenfunctions of the positive, self-adjoint compact integral operator$\mathcal{Q}_c^{\alpha}=c\mathcal{H}_c^{\alpha}\mathcal{H}_c^{\alpha}.$ TheCPSWFs play a central role in many applications such as the analysis of2d-radial signals. Moreover, a renewed interest on the CPSWFs instead ofFourier-Bessel basis is expected to follow from the potential applications inCryo-EM and that makes them attractive for steerable of principal componentanalysis(PCA). For this purpose, we give in this paper a precise non-asymptoticestimates for these eigenvalues, within the three main regions of the spectrumof $\mathcal{Q}_c^{\alpha}$ as well as these distributions in $(0,1).$Moreover, we describe a series expansion of CPSWFs with respect to thegeneralized Laguerre functions basis of $L^2(0,\infty)$ defined by$\psi_{n,\alpha}^a(x)=\sqrt{2}a^{\alpha+1}x^{\alpha+1/2}e^{-\frac{(ax)^2}{2}}\widetilde{L}_n^{\alpha}(a^2x^2)$,where $\widetilde{L}_n^{\alpha}$ is the normalised Laguerre polynomial.

Non-asymptotic behavior and the distribution of the spectrum of the finite Hankel transform operator