Polynomial Analogues of Prolate Spheroidal Wave Functions and Uncertainty

Abstract

Slepian, Landau, and Pollak used prolate spheroidal wave functions to demonstrate how nearly “time” and “bandlimited” a square-integrable function can be. In this note we show how their results extend easily to cover orthogonal polynomial expansions. In particular, we study how close a square-integrable function can come to being a polynomial of degree $ \leqq L$ and simultaneously to vanishing off some set $\mathcal{A}$.