Generalized number system and its applications (I)

Abstract

This is a further study of the work published in [2]. A generalized number system is established. The generalized functions denote the functions with generalized number system as their domain and range spaces which are different essentially from the Schwartz distributions. For such functions we define $(GNL)$ and $(G)$ integrals which can be considered as the generalization of the Lebesgue integral for real functions. Dirac $\delta$ function can be naturally represented by our generalized functions. This representation is more straightforward than the Schwartz distribution theory. Moreover, each distribution can be described by a generalized function in a natural way.