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

TL;DR

This paper investigates conditions under which finite sets of dilations generated by a one-parameter matrix Lie group are linearly dependent in L^p spaces, focusing on matrices with rationally related eigenvalues.

Contribution

It provides new criteria for linear dependence of dilated functions under one-parameter matrix Lie group actions with rational eigenvalues.

Findings

Identifies conditions for linear dependence in L^p spaces.

Analyzes the role of eigenvalue relations in dependence.

Establishes links between group structure and function dependence.

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 $1\leq p<\infty.$