An Interplay between Quadratic-Phase Fourier Transform and Octonion Algebra

Abstract

The quadratic-phase Fourier transform (QPFT) has emerged as a versatile five-parameter integral transform, unifying a wide range of unitary transformations, from the classical Fourier transform to the more recent special affine Fourier transform. However, its limitations become evident when applied to the precise representation of non-transient octonion-valued signals. To address this, we introduce the octonion quadratic-phase Fourier transform-a distinct integral transform specifically designed for such signals. In this study, we investigate its fundamental properties, including the energy-preserving relation and the inversion formula. Furthermore, we establish a set of uncertainty principles, such as Heisenberg’s and logarithmic uncertainty principles, providing deeper insights into the intricate interplay between octonion algebra and the quadratic-phase Fourier transform.

Keywords: Quadratic-phase Fourier transform; Quaternion Fourier transform; Octonion Fourier transform; Uncertainty principles.

2010 Mathematics Subject Classification. 42A38; 42C20; 11R52; 26D10; SDG9