Abstract
Let
𝔛
{\mathfrak{X}}
be a homogeneous tree and let
ℒ
{\mathcal{L}}
be the Laplace operator on
𝔛
{\mathfrak{X}}
. In this paper, we address problems of the following form: Suppose that
{
f
k
}
k
∈
ℤ
{\{f_{k}\}_{k\in\mathbb{Z}}}
is a doubly infinite sequence of functions in
𝔛
{\mathfrak{X}}
such that for all
k
∈
ℤ
{k\in\mathbb{Z}}
one has
ℒ
f
k
=
A
f
k
+
1
{\mathcal{L}f_{k}=Af_{k+1}}
and
∥
f
k
∥
≤
M
{\lVert f_{k}\rVert\leq M}
for some constants
A
∈
ℂ
{A\in\mathbb{C}}
,
M
>
0
{M>0}
and a suitable norm
∥
⋅
∥
{\lVert\,\cdot\,\rVert}
. From this hypothesis, we try to infer that
f
0
{f_{0}}
, and hence every
f
k
{f_{k}}
, is an eigenfunction of
ℒ
{\mathcal{L}}
. Moreover, we express
f
0
{f_{0}}
as the Poisson transform of functions defined on the boundary of
𝔛
{\mathfrak{X}}
.