A Theorem of Roe and Strichartz on homogeneous trees

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 $(d{f}_k/dx)=f_{k+1}$ 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.

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