Quaternion Hyperbolic Fourier Transforms and Uncertainty Principles

The present study introduces the two-sided and right-sided Quaternion Hyperbolic Fourier Transforms (QHFTs) for analyzing two-dimensional quaternion-valued signals defined in an open rectangle of the Euclidean plane endowed with a hyperbolic measure. The different forms of these transforms are defined by replacing the Euclidean plane waves with the corresponding hyperbolic plane waves in one dimension, giving the hyperbolic counterpart of the corresponding Euclidean Quaternion Fourier Transforms. Using hyperbolic geometry tools, we study the main operational and mapping properties of the QHFTs, such as linearity, shift, modulation, dilation, symmetry, inversion, and derivatives. Emphasis is placed on novel hyperbolic derivative and hyperbolic primitive concepts, which lead to the differentiation and integration properties of the QHFTs. We further prove the Riemann–Lebesgue Lemma and Parseval’s identity for the two-sided QHFT. Besides, we establish the Logarithmic, Heisenberg–Weyl, Donoho–Stark, and Benedicks’ uncertainty principles associated with the two-sided QHFT by invoking hyperbolic counterparts of the convolution, Pitt’s inequality, and the Poisson summation formula. This work is motivated by the potential applications of the QHFTs and the analysis of the corresponding hyperbolic quaternionic signals.