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 ${\psi }_{n,\alpha }^a(x)=\sqrt{2}{a}^{\alpha +1}{x}^{\alpha +1/2}{e}^{-\frac{(ax{)}^2}{2}}{\tilde{L}}_n^{\alpha }(a^2{x}^2)$, where ${\tilde{L}}_n^{\alpha }$ is the normalised Laguerre polynomial.