Linear Independence of a Finite Set of Dilations by a One-Parameter Matrix Lie Group

Abstract

Let $G=\{e^{tA}:t\in \mathbb{R}\}$ be a closed one-parameter subgroup of the general linear group of matrices of order $n$ acting on ${\mathbb{R}}^n$ by matrix-vector multiplications. We assume that all eigenvalues of $A$ are rationally related. We study conditions for which the set $f(e^{t_{1}A}.),.,f(e^{t_{m}A}.)$ is linearly dependent in $L^p({\mathbb{R}}^n)$ with $1\le p<\infty .$