Linear Dependency of Translations and Square Integrable Representations

TL;DR

This paper investigates the linear independence of translated functions in $L^2(G)$ for groups with specific representations, providing new results for groups like the affine, shearlet, and Weyl-Heisenberg groups.

Contribution

It establishes conditions for linear independence of translations in $L^2(G)$ for groups with square integrable, irreducible representations, extending understanding to specific groups and subgroup structures.

Findings

Results for the affine, shearlet, and Weyl-Heisenberg groups.

Conditions for linear independence in groups with abelian subgroups.

Insights into the structure of $L^2(G)$ under translation.

Abstract

Let $G$ be a locally compact group. We examine the problem of determining when nonzero functions in $L^2(G)$ have linearly independent translations. In particular, we establish some results for the case when $G$ has an irreducible, square integrable, unitary representation. We apply these results to the special cases of the affine group, the shearlet group and the Weyl-Heisenberg group. We also investigate the case when $G$ has an abelian, closed subgroup of finite index.