Differentiation and Multiplication by Functions

Abstract

Our motivations for distribution theory came largely from the limitations of the classical notion of differentiability. In this chapter we shall see that differentiation of distributions is indeed always possible. In addition we shall discuss multiplication. This operation on the other hand is not always defined unless one factor is smooth.

Differentiation of distributions and multiplication by smooth functions is defined in Section 3.1. As examples we discuss differentiation of functions with simple discontinuities which leads us to the Gauss-Green formula, and to Cauchy’s integral formula. As an application of the latter we digress to discuss boundary values in the distribution sense of analytic functions. As further illustration of multiplication and differentiation of distributions we discuss homogeneous distributions at some length in Section 3.2. Fundamental solutions of some classical second order differential operators are constructed in Section 3.3. In Section 3.4 finally we have collected some computations of integrals, particularly of Gaussian functions, which are needed in those constructions.