On the Prolate Spheroidal Wave Functions and Hardy’s Uncertainty Principle

We prove a weak version of Hardy’s uncertainty principle using properties of the prolate spheroidal wave functions. We describe the eigenvalues of the sum of a time limiting operator and a band limiting operator acting on L2(R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2(\mathbb {R})$$\end{document}. A weak version of Hardy’s uncertainty principle follows from the asymptotic behavior of the largest eigenvalue as the time limit and the band limit approach infinity. An asymptotic formula for this eigenvalue is obtained from its well-known counterpart for the prolate integral operator.