Kotani's Theorem for the Fourier Transform

TL;DR

This paper extends Kotani's 1991 theorem, providing conditions under which functions on integer lattices decay polynomially, based on properties of their Fourier transforms, with a proof inspired by Kotani's unpublished work.

Contribution

It offers a new proof and extension of Kotani's theorem relating Fourier transform behavior to function decay on lattices.

Findings

Extended Kotani's theorem with new sufficient conditions.

Provided a rigorous proof based on Kotani's original unpublished method.

Clarified the decay behavior of functions on ${f Z}^d$ from Fourier transform properties.

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 We prove an extension of Kotani's Theorem, based on Kotani's unpublished proof.