\section{Introduction}

Classical Hardy's uncertainty principle is formulated as follows.
\begin{theorem}
Let $a,b,M > 0$, and let $f$ be a measureable function on $\RR$ such that 
\begin{equation}
  |f(x)| \leq M \, e^{-ax^2/2} ,
\end{equation}
and
\begin{equation}
	|\hat{f}(\xi)| \leq M \, e^{-b\xi^2/2},
\end{equation}
for all $x,\xi \in \RR$. If $ab > 1$, then $f = 0$.
\end{theorem}
Several proofs of this classical theorem are known e.g.
\cite{hup_revisited, sharphardy, hardyoriginal, tao_blog}.  Typically,
they use methods of complex analysis, and rely on somewhat indirect
arguments. Our objective is to give a new and direct proof with
methods of real analysis, however only of the following weaker result.
\begin{theorem} \label{thm:hardyweak}
Let $a,b,M > 0$, and let $f$ be a measureable function on $\RR$ such that 
\begin{equation} \label{equ:boundtimedom}
  |f(x)| \leq M \, e^{-ax^2/2} 
\end{equation}
and
\begin{equation} \label{equ:boundfrequdom}
	|\hat{f}(\xi)| \leq M \, e^{-b\xi^2/2},
\end{equation}
for all $x,\xi \in \RR$. If $ab \geq 4$, then $f = 0$.
\end{theorem}

We prove this weak version of Hardy's uncertainty principle using
properties of the prolate spheroidal wave functions (PSWFs), which
appear e.g. in a solution of the concentration problem for bandlimited
functions \cite{slepian_conc}.  First, we describe the spectrum of the
sum of a time limiting operator and a band limiting operator acting on
$L^2(\mathbb{R})$. Specifically, we express the spectrum in terms of
the eigenvalues of the prolate integral operator.

Then we derive the weak version of Hardy's uncertainty principle from
the asymptotic behavior of the largest eigenvalue of the sum of the
time and band limiting operators 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.

Our approach reveals a relationship between Hardy's uncertainty
principle and the theory of bandlimited functions.
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