Colombeau’s Generalized Functions and Non-Standard Analysis

Abstract

Using some methods of Non-Standard Analysis we modify one of Colombeau’s classes of generalized functions. As a result we define a class Ê of so-called metafunctions which possesses all the good properties of Colombeau’s generalized functions, i.e. (i) Ê is an associative and commutative algebra over the system of so-called comptex meta-numbers ℂ̂(ii) Every meta-function has partial derivatives of any odrer (which are meta-functions again); (iii) Every meta-function is integrable on any compact set of ℝⁿ and the integral is a number from ℂ̂ (iv) Ê contains all the tempered distributions S’, i.e. S’ ⊂ Ê isomorphically with respect to all the linear operations (including the differentiation). Thus, within the class Ê the problem of multiplication of the tempered distributions is satisfactorily solved (every two distributions in S’ have a well-defined product in Ê). The crucial point is that ℂ̂ is a field in contrast to the system of Colombeau’s generalized numbers ℂ̂ which is a ring only ℂ̂ is the counterpart of ℂ̂ in Colombeau’s theory). In this way we simplify and improve slightly the properties of the integral and the notion of “values of the meta-functions”, as well as the properties of the whole class Ê itself if compared with the original Colombeau theory. And, what is maybe more important, we clarify the connection between Non-Standard Analysis and Colombeau's theory of new generalized function in the framework of which the problem of the multiplication of distributions was recently solved.