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[This post was typeset using a LaTeX to WordPress-HTML converter kindly provided to me by Luca Trevisan.]

Many properties of a (sufficiently nice) function ${f: {\mathbb R} \rightarrow {\mathbb C}}$ are reflected in its Fourier transform ${\hat f: {\mathbb R} \rightarrow {\mathbb C}}$, defined by the formula

$\displaystyle \hat f(\xi) := \int_{-\infty}^\infty f(x) e^{-2\pi i x \xi}\ dx. \ \ \ \ \ (1)$

For instance, decay properties of ${f}$ are reflected in smoothness properties of ${\hat f}$, as the following table shows:

 If ${f}$ is… then ${\hat f}$ is… and this relates to… Square-integrable square-integrable Plancherel’s theorem Absolutely integrable continuous Riemann-Lebesgue lemma Rapidly decreasing smooth theory of Schwartz functions Exponentially decreasing analytic in a strip Compactly supported entire and at most exponential growth Paley-Wiener theorem

Another important relationship between a function ${f}$ and its Fourier transform ${\hat f}$ is the uncertainty principle, which roughly asserts that if a function ${f}$ is highly localised in space, then its Fourier transform ${\hat f}$ must be widely dispersed in space, or to put it another way, ${f}$ and ${\hat f}$ cannot both decay too strongly at infinity (except of course in the degenerate case ${f=0}$). There are many ways to make this intuition precise. One of them is the Heisenberg uncertainty principle, which asserts that if we normalise

$\displaystyle \int_{{\mathbb R}} |f(x)|^2\ dx = \int_{\mathbb R} |\hat f(\xi)|^2\ d\xi = 1$

then we must have

$\displaystyle (\int_{\mathbb R} |x|^2 |f(x)|^2\ dx) \cdot (\int_{\mathbb R} |\xi|^2 |\hat f(\xi)|^2\ dx)\geq \frac{1}{(4\pi)^2}$

thus forcing at least one of ${f}$ or ${\hat f}$ to not be too concentrated near the origin. This principle can be proven (for sufficiently nice ${f}$, initially) by observing the integration by parts identity

$\displaystyle \langle xf, f' \rangle = \int_{\mathbb R} x f(x) \overline{f'(x)}\ dx = - \frac{1}{2} \int_{\mathbb R} |f(x)|^2\ dx$

and then using Cauchy-Schwarz and the Plancherel identity.

Another well known manifestation of the uncertainty principle is the fact that it is not possible for ${f}$ and ${\hat f}$ to both be compactly supported (unless of course they vanish entirely). This can be in fact be seen from the above table: if ${f}$ is compactly supported, then ${\hat f}$ is an entire function; but the zeroes of a non-zero entire function are isolated, yielding a contradiction unless ${f}$ vanishes. (Indeed, the table also shows that if one of ${f}$ and ${\hat f}$ is compactly supported, then the other cannot have exponential decay.)

On the other hand, we have the example of the Gaussian functions ${f(x) = e^{-\pi a x^2}}$, ${\hat f(\xi) = \frac{1}{\sqrt{a}} e^{-\pi \xi^2/a }}$, which both decay faster than exponentially. The classical Hardy uncertainty principle asserts, roughly speaking, that this is the fastest that ${f}$ and ${\hat f}$ can simultaneously decay:

Theorem 1 (Hardy uncertainty principle) Suppose that ${f}$ is a (measurable) function such that ${|f(x)| \leq C e^{-\pi a x^2 }}$ and ${|\hat f(\xi)| \leq C' e^{-\pi \xi^2/a}}$ for all ${x, \xi}$ and some ${C, C', a > 0}$. Then ${f(x)}$ is a scalar multiple of the gaussian ${e^{-\pi ax^2}}$.

This theorem is proven by complex-analytic methods, in particular the Phragmén-Lindelöf principle; for sake of completeness we give that proof below. But I was curious to see if there was a real-variable proof of the same theorem, avoiding the use of complex analysis. I was able to find the proof of a slightly weaker theorem:

Theorem 2 (Weak Hardy uncertainty principle) Suppose that ${f}$ is a non-zero (measurable) function such that ${|f(x)| \leq C e^{-\pi a x^2 }}$ and ${|\hat f(\xi)| \leq C' e^{-\pi b \xi^2}}$ for all ${x, \xi}$ and some ${C, C', a, b > 0}$. Then ${ab \leq C_0}$ for some absolute constant ${C_0}$.

Note that the correct value of ${C_0}$ should be ${1}$, as is implied by the true Hardy uncertainty principle. Despite the weaker statement, I thought the proof might still might be of interest as it is a little less “magical” than the complex-variable one, and so I am giving it below.