Wavelet like behavior of Slepian functions and their use in density estimation

Slepian functions (Prolate Spheroidal Wave Functions) are obtained by maximizing the energy of a sigma-bandlimited function (normalized with total energy 1) on a prescribed interval [-tau, tau]. The solution to this problem leads to an eigenvalue problem lambda f(t) = integral(tau)(-tau) {sin sigma(t - x)/pi(t - x)}f(x)d x, whose solutions, in turn, form an orthogonal sequence {phi(n)}. This sequence is a basis of the Paley-Wiener space B-sigma of sigma-bandlimited,functions. For sigma = pi, integer translates of the Slepian functions of order 0, {phi(0) (t - n)} form a Riesz basis of the same space. Furthermore, by using phi(0) as a scaling,function we can construct a wavelet theory based oil them. Two methods of density estimations thus naturally arise; one based on the orthogonal system {phi(n)) and the other on the scaling functions{phi(0)(t - n)}. The former gives more rapid convergence, while the latter avoids Gibbs phenomenon, is locally positive, and allows the use of thresholding methods. Both approaches exhibit a strong localization property.