Convolution and the Fourier Transform of Distributions

Abstract

As is the case for functions (Lesson 23), the Fourier transform interchanges convolution and multiplication of distributions. We wish to determine under what conditions the relations $\widehat {T*U} = \widehat T \cdot \widehat U$ and $$ are true. The first thing to notice is that one must be careful manipulating these relations, since the product of two distributions is not generally defined. We faced a similar problem with the convolution in the last lesson. There we were able to establish several conditions under which the convolution is well-defined and consistent with the convolution for functions.