# Extra invariance of principal shift invariant spaces and the Zak transform

## TL;DR

This paper characterizes when principal shift invariant spaces in L^2(R) are invariant under fractional translations, using the Zak transform, and extends the results to general LCA groups.

## Contribution

It provides a necessary and sufficient condition for invariance under fractional translations in principal shift invariant spaces, using the Zak transform, and generalizes to LCA groups.

## Key findings

- Characterization of invariance conditions via Zak transform
- Extension of results to LCA groups
- Framework for analyzing shift invariance in function spaces

## Abstract

We prove a necessary and sufficient condition for a principal shift invariant space of $L^2(\mathbb{R})$ to be invariant under translations by the subgroup $\frac{1}{N} \mathbb{Z}, N>1$. This condition is given in terms of the Zak transform of the group $\frac{1}{N} \mathbb{Z}.$ This result is extended to principal shift invariant spaces generated by a lattice in a general locally compact abelian (LCA) group.