Fourier, Gauss, and Heisenberg

Several weeks ago I wrote about the Fourier uncertainty principle which gives a lower bound on the product of the variance of a function f and the variance of its Fourier transform. This post expands on the earlier post by quoting some results from a recent paper [1].

Gaussian density

The earlier post said that the inequality in the Fourier uncertainty principle is exact when f is proportional to a Gaussian probability density. G. H. Hardy proved this result in 1933 in the form of the following theorem.

Let f be a square-integrable function on the real line and assume f and its Fourier transform satisfy the following bounds

\begin{align*} |f(x)| \leq& \,C \exp(-a|x|^2) \\ |\hat{f}(\xi)| \leq& \,C \exp(-b|\xi|^2\,) \\ \end{align*}

for some constant C. Then if ab > 1/4, then f = 0. And if ab = 1/4, f(x) = c exp(-ax²) for some constant c.

Let’s translate this into probability terms by setting

\begin{align*} a =& \,\frac{1}{2\sigma^2} \\ b =& \,\frac{1}{2\tau^2} \end{align*}

Now Hardy’s theorem says that if f is bounded by a multiple of a Gaussian density with variance σ² and its Fourier transform is bounded by a multiple of a Gaussian density with variance τ², then the product of the two variances is no greater than 1. And if the product of the variances equals 1, then f is a multiple of a Gaussian density with variance σ².

Heisenberg uncertainty

Theorem 3 in [1] says that if u(tx) is a solution to the free Schrödinger’s equation

\partial_t u = i \Delta u

then u at different points in time satisfies a theorem similar to Hardy’s theorem. In fact, the authors show that this theorem is equivalent to Hardy’s theorem.

Specifically, if u is a sufficiently smooth solution and

\begin{align*} |u(0,x)| \leq& \,C \exp(-\alpha|x|^2) \\ |u(T,x)| \leq& \,C \exp(-\beta|x|^2) \\ \end{align*}

then αβ > (4T)-2 implies u(t, x) = 0, and αβ = (4T)-2 implies

u(t,x) = c \exp(-(\alpha + i/(4T))|x|^2)

Related posts

[1] Aingeru Fernández-Bertolin and Eugenia Malinnikova. Dynamical versions of Hardy’s uncertainty principle: A survey. Bulletin of the American Mathematical Society. DOI: https://doi.org/10.1090/bull/1729

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