Kotani's Theorem for the Fourier Transform

In 1991, Shinichi Kotani proved a theorem giving a sufficient condition to conclude that a function $f(x)$ on ${\mathbb Z}^d$ decays like $|x|^{-(d-2)}$ for large $x$, assuming that its Fourier transform $\hat f(k)$ is such that $|k|^{2}\hat f(k)$ is well behaved for $k$ near zero. The proof was not published. We prove an extension of Kotani's Theorem, based on Kotani's unpublished proof.