Quaternion Hyperbolic Fourier-Type Transforms: An Overview

Ponente(s): Joao Pedro Leitao Da Cruz Morais

In this talk, we introduce the Hyperbolic Linear Canonical Transforms associated with two-dimensional quaternion-valued signals defined on an open rectangle of the Euclidean plane equipped with a hyperbolic measure. We refer to these as Quaternion Hyperbolic Linear Canonical Transforms (QHLCTs). Due to the non-commutative nature of quaternions, multiple forms of these transforms are possible. The definition of the QHLCTs is constructed by replacing classical Euclidean plane waves with hyperbolic relativistic plane waves in each dimension, in both the hyperbolic spatial and frequency domains. Here, two quaternion algebra generators assume the role of the imaginary unit, ensuring a separation between the two dimensions. We rigorously establish the fundamental properties of QHLCTs using tools from hyperbolic geometry and prove key results, including the Riemann-Lebesgue Lemma, the Plancherel and Parseval Theorems, and inversion formulas. Our analysis employs novel concepts of hyperbolic derivatives and hyperbolic primitives, which naturally lead to the differentiation and integration properties of the QHLCTs. We apply these results to formulate a quaternionic version of the Heisenberg uncertainty principle for the QHLCTs.