Compatibility Conditions and the Convolution of Functions and Generalized Functions

The paper is a review of certain existence theorems concerning the convolution of functions, distributions, and ultradistributions of Beurling type with supports satisfying suitable compatibility conditions. The fact that some conditions are essential for the existence of the convolution in the discussed spaces follows from earlier results and the proofs given at the end of this paper. In memory of Professor Jan Mikusiński on the 100th anniversary of his birthday 1. Introduction The convolution and its various generalizations play a very important role in the classical and abstract analysis as well as in other fields of mathematics, in particular in the theory of distributions (see [1–7]), ultradistributions (see [3, 4, 8–18]), hyperfunctions (see [19–21]), and other generalized functions considered for various spaces, subspaces, and approaches. The notion of convolution is a starting point in algebraic approaches to certain generalized functions: the convolution algebra of continuous functions on extends, due to the Titchmarsh theorem, to the field of Mikusiński operators (see [22–25]) which are generalized functions of another type than Schwartz distributions, while Boehmians stand for a common generalization of regular Mikusiński operators of Boehme (see [26]), Schwartz distributions, and other classes of generalized functions on the real line (see [27–31]). An important part of investigations connected with the convolution is the study of convolution operators and convolution semigroups for various spaces of functions and generalized functions (see, e.g., [32–37]). Therefore the problems concerning the existence of the convolution in various spaces of functions and generalized functions are crucial. The theory developed by Colombeau (see [38]; see also [39]) and his followers (see, e.g., [40–43]) has led to constructions of algebras of new generalized functions related to the distributions and other classical generalized functions due to certain quotient procedures; consequently the algebras of new generalized functions are closed with respect to multiplication as well as other nonlinear operations. However the problem of existence of the product and the convolution of distributions and other generalized functions in the standard sense, without using Colombeau’s approach, remains important. We will analyse in this paper the existence of the convolution of distributions and tempered distributions on meant in the classical sense of general equivalent definitions introduced independently by several authors (see [2, 44–50]). Also the existence of the