The quaternion Fourier and wavelet transforms on spaces of functions and distributions (Q783078)

Irish mathematician Sir W. R. Hamilton was the first to introduce the quaternion algebra in 1843. The quaternions are an extension of complex numbers to a four-dimensional associative non-commutative algebra. In this paper, the right-hand sided quaternion Fourier transform, which is a non-trivial generalization of the real and complex Fourier transform, is studied on spaces of test functions and distributions. The continuous quaternion wavelet transform of periodic functions is also defined and its quaternion Fourier representation form is established. The Plancherel and inversion formulas for the continuous quaternion wavelet transform are established.