Calculus for Distributions

Abstract

The basic operations of calculus are introduced in such a way that they are compatible with the rules of calculus for functions. This is achieved through the duality method. First differentiation is introduced and it turns out that all distributions are infinitely often differentiable. Next multiplication of distributions with $\mathcal{C}^{\infty}$ functions is defined and it follows that the product rule of differentiation holds. Tempered distributions can only be multiplied by $\mathcal{C}^{\infty}$ functions which together with all their derivatives do not grow faster that a polynomial. Transformation of variables can be defined for distributions if they are effected by $\mathcal{C}^{\infty}$ functions and then the chain rule of differentiation holds. Several examples illustrate these definitions. In particular the general form of distributions with support in a point is determined.