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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.

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