Theory of hypernumbers and extrafunctions: Functional spaces and differentiation

Abstract

The theory of hypernumbers and extrafunctions is a novel approach in functional analysis aimed at problems of mathematical and computational physics. The new technique allows operations with divergent integrals and series and makes it possible to distinct different kinds of convergence and divergence. Although, it resembles nonstandard analysis, there are several distinctions between these theories. For example, while nonstandard analysis changes spaces of real and complex numbers by injecting into them infinitely small numbers and other nonstandard entities, the theory of extrafunctions does not change the inner structure of spaces of real and complex numbers, but adds to them infinitely big and oscillating numbers as external objects. In this paper, we consider a simplified version of hypernumbers, but a more general version of extrafunctions and their extraderivatives in comparison with previous works.