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AIP conference proceedings

Octonion linear canonical transform: The uncertainty principle and sharp inequalities

Manish Kumar, Bhawna

Published Jan 1, 2026• 0 citations
Publisher

Abstract

This paper established Sharp-Pitt’s inequality using the Octonion Linear Canonical Transform (OCLCT), which we subsequently used to demonstrate the logarithmic uncertainty principle. Further, Sharp-Hausdorff-Young’s inequality is also ob-tained. Some examples of important basic functions (or signals) for OCLCT are constructed, and their particular cases are analyzed through graphical visualization, including interpretation.

DOI
10.1063/5.0348304

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This paper established Sharp-Pitt’s inequality using the Octonion Linear Canonical Transform (OCLCT), which we subsequently used to demonstrate the logarithmic uncertainty principle. Further, Sharp-Hausdorff-Young’s inequality is also ob-tained. Some examples of important basic functions (or signals) for OCLCT are constructed, and their particular cases are analyzed through graphical visualization, including interpretation.
The uncertainty principle plays an important role in mathematics, physics, signal processing, and so on. Firstly, based on definition of the linear canonical transform (LCT) and the traditional Pitt's inequality, one novel Pitt's inequality in the LCT domains is obtained, which is connected with the LCT parameters and Then one novel logarithmic uncertainty principle is derived from this novel Pitt's inequality in the LCT domains, which is associated with parameters of the two LCTs. Secondly, from the relation between the original function and LCT, one entropic uncertainty principle and one Heisenberg's uncertainty principle in the LCT domains are derived, which are associated with the LCT parameters and The reason why the three lower bounds are only associated with LCT parameters and and independent of and is presented. The results show it is possible that the bounds tend to zeros.
Uncertainty principle plays an important role in multiple fields such as physics, mathematics, signal processing, etc. The linear canonical transform (LCT) has been used widely in optics and information processing and so on. In this paper, a few novel uncertainty inequalities on Fisher information associated with linear canonical transform are deduced. These newly deduced uncertainty relations not only introduce new physical interpretation in signal processing, but also build the relations between the uncertainty lower bounds and the LCT transform parameters a, b, c and d for the first time, which give us the new ideas for the analysis and potential applications. In addition, these new uncertainty inequalities have sharper and tighter bounds which are the generalized versions of the traditional counterparts. Furthermore, some numeric examples are given to demonstrate the efficiency of these newly deduced uncertainty inequalities.
The offset linear canonical transform (OLCT) provides a more general framework for a number of well known linear integral transforms in signal processing and optics, such as Fourier transform, fractional Fourier transform, linear canonical transform. In this paper, to characterize simultaneous localization of a signal and its OLCT, we generalize some different uncertainty principles (UPs), including Nazarov's UP, Hardy's UP, Beurling's UP, logarithmic UP and entropic UP, which have already been well studied in the Fourier transform domain over the last few decades, to the OLCT domain in a broader sense.