An Algebraic Approach to Distribution Theories

Abstract

In this note Mikusiński’s idea of convolution quotients is generalized in two directions simultaneously:

• – The ring R is not merely acting on itself but acts also on a separate vector space V.

• – The ring R need not be commutative.

This approach makes it possible to bring many theories of generalized functions under the same viewpoint. So e.g. Schwartz’ tempered distribution space, many (all?) distribution spaces of Gelfand-Šilov and Mikusiński’s space of convolution quitients (cf. the Examples) are all very special examples of the construction that we present here. The authors expect that many more locally convex topological vector spaces can be brought under the same viewpoint. The connection with (non-commutative) harmonic analysis, especially the case that R is a (subset of) the convolution algebra of a Lie group is now being studied by the second author. The algebra of this paper is inspired by Ore’s construction of non-commutative fields. See [5], p. 119.