Relationship between the cayley-dickson fourier transform and the hartley transform of multidimensional real signals

The paper shows relations between two different frequency representations of n-dimensional signals (n = 1, 2, 3): the Cayley-Dickson Fourier transform and the Hartley transform. The formulas relating the n-D complex Fourier transform and the n-D Hartley transform are presented. New formulas relating the Quaternion and Octonion Fourier transforms and respectively the 2-D and 3-D Hartley transforms are developed. The paper is illustrated with examples of Hartley transforms of 1-D, 2-D and 3-D Gaussian signals.