Elements of the Theory of Generalized Functions

Abstract

In this chapter, we present the principles of the theory of generalized functions of the Sobolev-Schwartz-type. We consider classical spaces of test functions D(ℝ^N) and S(ℝ^N), i.e., the linear spaces of finite and rapidly decreasing smooth functions equipped with proper convergences. We study the properties of generalized functions from the spaces D′(ℝ^N) and S′(ℝ^N) of linear continuous functionals on D(ℝ^N) and S(ℝ^N), respectively. The theory of generalized functions can also he constructed in a more general way by using the theory of linear topological spaces. This approach is considered in Sections 14.2 and 14.4.