On the prolate spheroidal wave functions and Hardy's uncertainty principle

TL;DR

This paper establishes a weak form of Hardy's uncertainty principle by analyzing the eigenvalues of operators involving prolate spheroidal wave functions, revealing their asymptotic behavior as limits grow large.

Contribution

It introduces a novel connection between Hardy's uncertainty principle and the spectral properties of prolate spheroidal wave functions, providing new asymptotic formulas.

Findings

Eigenvalues of combined time and band limiting operators are characterized.

Asymptotic behavior of the largest eigenvalue is derived.

A weak version of Hardy's uncertainty principle is proved.

Abstract

We prove a weak version of Hardy's uncertainty principle using properties of the prolate spheroidal wave functions (PSWFs). We describe the eigenvalues of the sum of a time limiting operator and a band limiting operator acting on L2(R). 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.