The Zak transform on strongly proper $G$-spaces and its applications

TL;DR

This paper generalizes the Zak transform to a broad class of locally compact G-spaces, unifying previous versions and establishing its invariance and isomorphism properties, with applications in physics.

Contribution

It introduces a unified framework for the Zak transform on G-spaces, extending its applicability and theoretical understanding.

Findings

Zak transform is a Hilbert space isomorphism on G-spaces.

Unifies previous Zak transform generalizations.

Provides explicit formulas using Weil and Poisson summation.

Abstract

The Zak transform on $\mathbb{R}^d$ is an important tool in condensed matter physics, signal processing, time-frequency analysis, and harmonic analysis in general. This article introduces a generalization of the Zak transform to a class of locally compact $G$-spaces, where $G$ is either a locally compact abelian or a second countable unimodular type I group. This framework unifies previously proposed generalizations of the Zak transform. It is shown that the Zak transform has invariance properties analog to the classic case and is a Hilbert space isomorphism between the space of $L^2$-functions and a direct integral of Hilbert spaces that is explicitly determined via a Weil formula for $G$-spaces and a Poisson summation formula for compact subgroups. Some applications in physics are outlined.