Kotani's Theorem for the Fourier Transform

Abstract

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 We prove an extension of Kotani's Theorem, based on Kotani's unpublished proof.

Version 2 with new title consists of Section 2 of Version 1. Section 3 of Version 1 (lace expansion) is now the basis of an independent paper. 6 pages