Wavelet Like Behavior of Slepian Functions and Their Use in Density Estimation

Abstract

Slepian functions (Prolate Spheroidal Wave Functions) are obtained by maximizing the energy of a σ-bandlimited function (normalized with total energy 1) on a prescribed interval [−τ, τ]. The solution to this problem leads to an eigenvalue problem λf(t) = {sinσ(t − x)/π(t − x)}f(x)dx, whose solutions, in turn, form an orthogonal sequence {ϕ n }. This sequence is a basis of the Paley-Wiener space B σ of σ-bandlimited functions. For σ = π, integer translates of the Slepian functions of order 0, {ϕ0(t − n)} form a Riesz basis of the same space. Furthermore, by using ϕ0 as a scaling function we can construct a wavelet theory based on them. Two methods of density estimations thus naturally arise; one based on the orthogonal system {ϕ n } and the other on the scaling functions {ϕ0(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.