Mellin transforms over Cayley-Dickson algebras

A study was conducted to analyze new types of direct and inverse Mellin-type transforms over the Cayley-Dickson-algebras, in particular, over the quaternion skew field and the octonion algebra. The study considered the auxiliary function, which naturally arises when dealing with spherical coordinates in the basis of generators of the algebra and in considering iterated exponential functions. A function is called the function-original of the noncommunicative Mellin transform over the Cayley-Dickson-algebra, if it satisfies the Hölder condition. The convergence of the integrals in the study implies the (super)differentiability of the function. The study also found that it is sufficient to prove that the required equality retains its form under the action of the automorphism on its left- and right-hand sides.