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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 are reflected in its Fourier transform , defined by the formula
For instance, decay properties of are reflected in smoothness properties of , as the following table shows:
|If is…||then is…||and this relates to…|
|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 and its Fourier transform is the uncertainty principle, which roughly asserts that if a function is highly localised in space, then its Fourier transform must be widely dispersed in space, or to put it another way, and cannot both decay too strongly at infinity (except of course in the degenerate case ). There are many ways to make this intuition precise. One of them is the Heisenberg uncertainty principle, which asserts that if we normalise
then we must have
thus forcing at least one of or to not be too concentrated near the origin. This principle can be proven (for sufficiently nice , initially) by observing the integration by parts identity
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 and to both be compactly supported (unless of course they vanish entirely). This can be in fact be seen from the above table: if is compactly supported, then is an entire function; but the zeroes of a non-zero entire function are isolated, yielding a contradiction unless vanishes. (Indeed, the table also shows that if one of and is compactly supported, then the other cannot have exponential decay.)
On the other hand, we have the example of the Gaussian functions , , which both decay faster than exponentially. The classical Hardy uncertainty principle asserts, roughly speaking, that this is the fastest that and can simultaneously decay:
Theorem 1 (Hardy uncertainty principle) Suppose that is a (measurable) function such that and for all and some . Then is a scalar multiple of the gaussian .
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:
Note that the correct value of should be , 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.