Prolate spheroidal wave functions

Summary

The prolate spheroidal wave functions are the bounded solutions on $[-1, 1]$ of a second-order differential equation that arises from separating variables in prolate spheroidal coordinates. D. Slepian and H. O. Pollak found that the same functions are eigenfunctions of a time and band limiting operator. This operator first restricts a function to $[-1, 1]$ and then removes all frequencies outside a fixed band. So they are the band-limited functions that are most concentrated on an interval. They answer the question of how much of its energy a band-limited signal can keep in a given time window. They are used in signal processing and spectral estimation. See the definition.