The two-sided Laplace transformation over the Cayley-Dickson algebras and its applications

Abstract

This paper is devoted to the noncommutative version of the Laplace transformation. New types of direct and inverse transformations of the Laplace type over general Cayley-Dickson algebras, in particular, the skew field of quaternions and the octonion algebra are investigated. Examples are given. Theorems about properties of such transformations and also theorems about images and originals in conjunction with the operations of multiplication, differentiation, integration, convolution, shift, and homothety are proved. Applications are given to a solution of differential equations over the Cayley-Dickson algebras. © 2009 Springer Science+Business Media, Inc.