# Approximation by group invariant subspaces

## TL;DR

This paper characterizes and constructs group invariant subspaces of L^2(R) under non-commutative group actions, with applications to signal processing involving rigid movements.

## Contribution

It provides a complete characterization of b3-invariant subspaces using new range functions and constructs Parseval frames, extending the theory to non-commutative group actions.

## Key findings

- Complete characterization of b3-invariant subspaces via range functions.
- Construction of Parseval frames for these subspaces.
- Application to optimal approximation of functional data in signal processing.

## Abstract

In this article we study the structure of $\Gamma$-invariant spaces of $L^2(\bf R)$. Here $\bf R$ is a second countable LCA group. The invariance is with respect to the action of $\Gamma$, a non commutative group in the form of a semidirect product of a discrete cocompact subgroup of $\bf R$ and a group of automorphisms. This class includes in particular most of the crystallographic groups. We obtain a complete characterization of $\Gamma$-invariant subspaces in terms of range functions associated to shift-invariant spaces. We also define a new notion of range function adapted to the $\Gamma$-invariance and construct Parseval frames of orbits of some elements in the subspace, under the group action. These results are then applied to prove the existence and construction of a $\Gamma$-invariant subspace that best approximates a set of functional data in $L^2(\bf R)$. This is very relevant in applications since in the euclidean case, $\Gamma$-invariant subspaces are invariant under rigid movements, a very sought feature in models for signal processing.