The Mapping of the Convolution

Abstract

In Chapters 7, 8, and 9, we studied the effect of operations involving one function such as, for instance, differentiation. In this Chapter, we shall investigate operations which involve more than one function, like sum or product of functions. The formula for the sum of functions

$${\cal L}\left\{ {{f_1} + {f_2}} \right\}\,\, = \,\,{\cal L}\left\{ {{f_1}} \right\}\, + {\cal L}\left\{ {{f_2}} \right\}$$

is immediately obvious. However, the image of the product of two original functions, f₁ · f₂ say, is quite complicated, thus necessitating that its study be deferred to Chapter 31.