[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.
— 1. The complex-variable proof —
We first give the complex-variable proof. By dilating by (and contracting by ) we may normalise . By multiplying by a small constant we may also normalise .
The super-exponential decay of allows us to extend the Fourier transform to the complex plane, thus
for all . We may differentiate under the integral sign and verify that is entire. Taking absolute values, we obtain the upper bound
for all . We conclude that the entire function
is bounded in magnitude by on the imaginary axis; also, by hypothesis on , we also know that is bounded in magnitude by on the real axis. Formally applying the Phragmen-Lindelöf principle (or maximum modulus principle), we conclude that is bounded on the entire complex plane, which by Liouville’s theorem implies that is constant, and the claim follows.
Now let’s go back and justify the Phragmén-Lindelöf argument. Strictly speaking, Phragmén-Lindelöf does not apply, since it requires exponential growth on the function , whereas we have quadratic-exponential growth here. But we can tweak a bit to solve this problem. Firstly, we pick and work on the sector
Using (2) we have
Thus, if , and is sufficiently close to depending on , the function is bounded in magnitude by on the boundary of . Then, for any sufficiently small , (using the standard branch of on ) is also bounded in magnitude by on this boundary, and goes to zero at infinity in the interior of , so is bounded by in that interior by the maximum modulus principle. Sending , and then , and then , we obtain bounded in magnitude by on the upper right quadrant. Similar arguments work for the other quadrants, and the claim follows.
— 2. The real-variable proof —
Now we turn to the real-variable proof of Theorem 2, which is based on the fact that polynomials of controlled degree do not resemble rapidly decreasing functions.
Rather than use complex analyticity , we will rely instead on a different relationship between the decay of and the regularity of , as follows:
Lemma 3 (Derivative bound) Suppose that for all , and some . Then is smooth, and furthermore one has the bound for all and every even integer .
Proof: The smoothness of follows from the rapid decrease of . To get the bound, we differentiate under the integral sign (one can easily check that this is justified) to obtain
and thus by the triangle inequality for integrals (and the hypothesis that is even)
On the other hand, by differentiating the Fourier analytic identity
times at , we obtain
expanding out using Taylor series we conclude that
Using Stirling’s formula , we conclude in particular that
for all large even integers (where the decay of can depend on ).
We can combine (3) with Taylor’s theorem with remainder, to conclude that on any interval , we have an approximation
Now we apply a useful bound.
Lemma 4 (Doubling bound) Let be a polynomial of degree at most for some , let be an interval, and suppose that for all and some . Then for any we have the bound for all and for some absolute constant .
Proof: By translating we may take ; by dilating we may take . By dividing by , we may normalise . Thus we have for all , and the aim is now to show that for all .
Consider the trigonometric polynomial . By de Moivre’s formula, this function is a linear combination of for . By Fourier analysis, we can thus write , where
Since is bounded in magnitude by , we conclude that is bounded in magnitude by . Next, we use de Moivre’s formula again to expand as a linear combination of and , with coefficients of size ; expanding further as , we see that is a polynomial in with coefficients . Putting all this together, we conclude that the coefficients of are all of size , and the claim follows. ◻
Remark 1 One can get slightly sharper results by using the theory of Chebyshev polynomials. (Is the best bound for known? I do not know the recent literature on this subject. I think though that even the sharpest bound for would not fully recover the sharp Hardy uncertainty principle, at least with the argument given here.)
for , and some polynomial of degree . In particular, we have
for . Applying Lemma 4, we conclude that
for . Applying (4) again we conclude that
for . If we pick for a sufficiently small absolute constant , we conclude that
(say) for . If for large enough , the right-hand side goes to zero as (which also implies ), and we conclude that (and hence ) vanishes identically.