Hypercomplex Function Theory and Representation of Distributions

Abstract

Publisher Summary This chapter discusses the hypercomplex function theory and representation of distributions. A representation theory for distributions by means of a class of hypercomplex functions that is a rather natural extension to the space of the class of holomorphic functions of one complex variable is presented. An excellent survey on the approach of distributions and ultra-distributions by holomorphic functions has been discussed in the chapter. It is shown that the relationship between the distribution and holomorphic function spaces is topological. It is found that in the sequel, each of the function spaces, equipped with its natural topology are considered as a left A-module and the set of bounded left A-linear functionals on it are denoted, respectively. The construction of primitives and continuous extensions of monogenic functions is explained. The relationship between the distributional extension properties and distributional boundary values of monogenic functions is discussed. The space of monogenic functions that have distributional boundary values is also discussed in the chapter.