On the polyanalytic short-time Fourier transform in the quaternionic setting

TL;DR

This paper develops a quaternionic short-time Fourier transform using Hermite functions, establishing new properties like Moyal and reconstruction formulas, and extends prior work on Gaussian-windowed QSTFT within the framework of slice polyanalytic functions.

Contribution

It introduces a novel QSTFT based on Hermite functions and slice polyanalytic function theory, providing foundational properties and extending previous Gaussian-based results.

Findings

Established a Moyal formula for the QSTFT.

Proved a reconstruction formula for the QSTFT.

Derived Lieb's uncertainty principle in the quaternionic setting.

Abstract

In this paper, we consider a quaternionic short-time Fourier transform (QSTFT) with normalized Hermite functions as windows. It turns out that such a transform is based on the recent theory of slice polyanalytic functions on quaternions. Indeed, we will use the notions of true and full slice polyanalytic Fock spaces and Segal-Bargmann transforms. We prove new properties of this QSTFT including a Moyal formula, a reconstruction formula and a Lieb's uncertainty principle. These results extend a recent paper of the authors which studies a QSTFT having a Gaussian function as a window.