Boehmians and generalized functions (Q1107058)

This paper makes a unity with the two ones published previously by \textit{J. Mikusiński} [C. R. Acad. Sci., Paris, Sér. I 293, 463-464 (1981; Zbl 0495.44006)] and the author [Japan J. Math. 9, 159-179 (1983; Zbl 0524.44005)] and completes them. The most interesting results are given in the third part of the paper which is devoted to the convergence of Boehmians and in the last part in which the author discusses the connection between Boehmians and other generalized functions, used at the most. It is interesting that Boehmians contain all distributions. We cite only one theorem. Let G be a linear space, S subspace of G and \(\Delta\) be a family of delta sequences, then by \(B(G,\Delta)\) we denote the set of Boehmians and \(B_ F(G,\Delta)=\{f\in B(G,\Delta),f*\phi \in G,\phi \in F\},\) F is a subsemigroup of S. Theorem: ``Let F be a subalgebra of the convolution algebra \({\mathcal D}\). If \(F^ N\cap \Delta_ c\neq \emptyset\), then the completion of C with respect to F- convergence is isomorphic to \(B_ F(C,\Delta_ c)\).