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 $\mathrm{\mid }x{\mathrm{\mid }}^{-(d-2)}$ for large $x$, assuming that its Fourier transform $\widehat{f}(k)$ is such that $\mathrm{\mid }k{\mathrm{\mid }}^2\widehat{f}(k)$ 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 We prove an extension of Kotani’s Theorem, based on Kotani’s unpublished proof.