Subspaces of $L^2(\mathbb{R}^n)$ Invariant Under Shifts by a Crystal Group

TL;DR

This paper characterizes the structure of subspaces in $L^2(R^n)$ that remain invariant under shifts by a crystal group, extending understanding of symmetry-invariant function spaces in harmonic analysis.

Contribution

It provides a complete characterization of $Gamma$-shift invariant subspaces of $L^2(R^n)$ for crystal groups, generalizing previous results to higher dimensions.

Findings

Characterization of $Gamma$-shift invariant subspaces

Extension of shift-invariance concepts to crystal groups

Framework applicable to harmonic analysis and signal processing

Abstract

For a crystal group $\Gamma$ in dimension $n$, a closed subspace $\mathcal{V}$ of $L^2(\mathbb{R}^n)$ is called $\Gamma$--shift invariant if, for every $f\in\mathcal{V}$, the shifts of $f$ by every element of $\Gamma$ also belong to $\mathcal{V}$. The main purpose of this paper is to provide a characterization of the $\Gamma$--shift invariant closed subspaces of $L^2(\mathbb{R}^n)$.