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

Abstract

For a fixed reals $c>0$, $a>0$ and $\alpha>-\frac{1}{2}$, the circular prolate spheroidal wave functions (CPSWFs) or 2d-Slepian functions as some authors call it, are the eigenfunctions of the finite Hankel transform operator, denoted by $\mathcal{H}_c^{\alpha}$, which is the integral operator defined on $L^2(0,1)$ with kernel $H_c^{\alpha}(x,y)=\sqrt{cxy}J_{\alpha}(cxy)$. Also, they are the eigenfunctions of the positive, self-adjoint compact integral operator $\mathcal{Q}_c^{\alpha}=c\mathcal{H}_c^{\alpha}\mathcal{H}_c^{\alpha}.$ The CPSWFs play a central role in many applications such as the analysis of 2d-radial signals. Moreover, a renewed interest on the CPSWFs instead of Fourier-Bessel basis is expected to follow from the potential applications in Cryo-EM and that makes them attractive for steerable of principal component analysis(PCA). For this purpose, we give in this paper a precise non-asymptotic estimates for these eigenvalues, within the three main regions of the spectrum of $\mathcal{Q}_c^{\alpha}$ as well as these distributions in $(0,1).$ Moreover, we describe a series expansion of CPSWFs with respect to the generalized 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.