Fourier analysis of sequences over a composition algebra of the real number field

To analyze the structure of a set of perfect sequences over a composition algebra of the real number field, transforms of a set of sequences similar to DFT (discrete Fourier transform) are introduced. Discrete cosine transform, discrete sine transform and generalized discrete Fourier transform (GDFT) of the sequences are defined and the fundamental properties of these transforms are proved. We show that GDFT is bijective and that there exists a relationship between these transforms and a convolution of sequences. Applying these properties to the set of perfect sequences, a parameterization theorem of such sequences is obtained.