Generalized Uncertainty Principles associated with the Quaternionic Offset Linear Canonical Transform

TL;DR

This paper introduces the quaternionic offset linear canonical transform (QOLCT), explores its properties, and extends various uncertainty principles to this new transform, broadening the mathematical framework for quaternionic signal analysis.

Contribution

It defines the QOLCT, establishes its relationship with the quaternion Fourier transform, and generalizes several uncertainty principles to this transform domain.

Findings

Derived the relationship between QOLCT and QFT

Proved the Plancherel formula for QOLCT

Generalized multiple uncertainty principles to QOLCT domain

Abstract

The quaternionic offset linear canonical transform (QOLCT) can be thought as a generalization of the quaternionic linear canonical transform (QLCT). In this paper we define the QOLCT, we derive the relationship between the QOLCT and the quaternion Fourier transform (QFT). Based on this fact we prove the Plancherel formula, and some properties related to the QOLCT, then we generalize some different uncertainty principles (UPs), including Heisenberg-Weyls UP, Hardys UP, Beurlings UP, and logarithmic UP to the QOLCT domain in a broader sense