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

For fixed reals c > 0, a > 0 and alpha > -1/2, the circular prolate spheroidal wave functions (CPSWFs) or 2d-Slepian functions are the eigenfunctions of the finite Hankel transform operator, denoted by H-c(alpha), which is the integral operator defined on L-2(0, 1) with kernel H-c(alpha)(x, y) = root cxyJ(alpha) (cxy). Also, they are the eigenfunctions of the positive, self-adjoint compact integral operator Q(c)(alpha) = cH(c)(alpha)H(c)(alpha). The CPSWFs play a central role in many applications such as the analysis of 2d-radial signals. Moreover, a renewed interest in 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 precise non-asymptotic estimates for the eigenvalues of Q(c)(alpha), within the three main regions of the spectrum of Q(c)(alpha). Moreover, we describe a series expansion of CPSWFs with respect to the generalized Laguerre functions basis of L-2(0, infinity) defined by psi(a)(n,alpha) (x) = root 2a(alpha+1) x(alpha+1/2) e ((ax)2/2) (L) over tilde (alpha)(n) (a(2)x(2)), where (L) over tilde (alpha)(n) is the normalized Laguerre polynomial.