Asymptotic behaviour of the quaternion linear canonical transform and the Bochner-Minlos theorem

There have been numerous proposals in the literature to generalize the classical Fourier transform by making use of the Hamiltonian quaternion algebra. The present paper reviews the quaternion linear canonical transform (QLCT) which is a generalization of the quaternion Fourier transform and it studies a number of its properties. In the first part of this paper, we establish a generalized Riemann-Lebesgue lemma for the (right-sided) QLCT. This lemma prescribes the asymptotic behaviour of the QLCT extending and refining the classical Riemann-Lebesgue lemma for the Fourier transform of 2D quaternion signals. We then introduce the QLCT of a probability measure, and we study some of its basic properties such as linearity, reconstruction formula, continuity, boundedness, and positivity. Finally, we extend the classical Bochner-Minlos theorem to the QLCT setting showing the applicability of our approach.