Some New Properties Using Quaternionic Fourier–Mellin Transform on the Space \(L^{2}(G, {\mathbb {H}})\)

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 \(G={\mathbb {R}}_{+}^{*}\times {\mathbb {S}}^{ 1}\) $G=\mathbb{R}_{+}^{\ast }\times {\mathbb{S}}^1$ and \({\mathbb {S}}^{ 1}\) ${\mathbb{S}}^1$ denotes the unit circle of the plane \({\mathbb {R}}^{2}\) ${\mathbb{R}}^2$ . 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 \(L^{2}(G, {\mathbb {H}})\) $L^2(G,\mathbb{H})$ .