# A Theorem of Roe and Strichartz on homogeneous trees

## TL;DR

This paper extends Roe and Strichartz's classical results on bounded function sequences satisfying specific differential relations from Euclidean spaces to the setting of homogeneous trees, revealing analogous structural properties.

## Contribution

It provides a new theorem characterizing functions on homogeneous trees that satisfy a difference relation similar to the differential relation in Euclidean spaces.

## Key findings

- Established a Roe-Strichartz type theorem for homogeneous trees.
- Identified the form of functions satisfying the difference relation on trees.
- Extended classical Euclidean results to a discrete, non-Euclidean setting.

## Abstract

In 1980, J. Roe proved that if $\{f_{k}\}_{k\in\mathbb{Z}}$ is doubly infinite sequence of functions in $\mathbb{R}$ which is uniformly bounded and satisfies $(df_{k}/dx)=f_{k+1}$ for all $k\in\mathbb{Z}$ then $f_{0}(x)=a\sin(x+\theta)$ for some $a,\theta\in\mathbb{R}$. Later in 1993 Strichartz suitably extended the above result to $\mathbb{R}^n$. In this article we prove a version of their result for homogeneous trees.