Some New Properties Using Quaternionic Fourier–Mellin Transform on the Space L2(G,H)

In this present work, by using the Quaternionic Fourier–Mellin transform and a new translation operator, we give some characterizations of quaternion-valued functions satisfying certain Lipschitz conditions on G , where G = R + ∗ × S 1 and S 1 denotes the unit circle of the plane R 2 . Moreover, in the second main result of this paper, after defining the Fourier–Mellin K-functional and the Fourier–Mellin modulus of smoothness, we prove their equivalence on the spaces L 2 ( G , H ) .