  We prove that the density function of the gradient of a sufficiently smooth
function $S : \Omega \subset \mathbb{R}^d \rightarrow \mathbb{R}$, obtained via
a random variable transformation of a uniformly distributed random variable, is
increasingly closely approximated by the normalized power spectrum of
$\phi=\exp\left(\frac{iS}{\tau}\right)$ as the free parameter $\tau \rightarrow
0$. The result is shown using the stationary phase approximation and standard
integration techniques and requires proper ordering of limits. We highlight a
relationship with the well-known characteristic function approach to density
estimation, and detail why our result is distinct from this approach.
