On the quaternionic short-time Fourier and Segal-Bargmann transforms

TL;DR

This paper explores a quaternionic short-time Fourier transform based on the slice hyperholomorphic Segal-Bargmann transform, establishing its fundamental properties, formulas, and uncertainty principles.

Contribution

It introduces a novel quaternionic STFT framework derived from the slice hyperholomorphic Segal-Bargmann transform, with comprehensive theoretical analysis.

Findings

Established Moyal formula for quaternionic STFT

Proved reconstruction formula in quaternionic setting

Derived Lieb's uncertainty principle for QSTFT

Abstract

In this paper, we study a special one dimensional quaternion short-time Fourier transform (QSTFT). Its construction is based on the slice hyperholomorphic Segal-Bargmann transform. We discuss some basic properties and prove different results on the QSTFT such as Moyal formula, reconstruction formula and Lieb's uncertainty principle. We provide also the reproducing kernel associated to the Gabor space considered in this setting.