Summary: A space of original functions which are continuous with respect to \(n\) variables and discrete with respect to \(m\) variables is presented. A multiple hybrid Laplace and \(z\) type transformation is defined on this set. Its main properties are studied, including linearity, time-delay, translation, differentiation and difference of the original, differentiation of the image. Other theorems such as integration and sum of the original, convolution, initial and final values etc. will be presented in a subsequent paper, as well as some methods to determine the originals.
These properties will be used to solve multiple differential-difference and multiple integral equations and to obtain the frequency-domain representations of multidimensional hybrid control systems.