Convolution of Distributions

The importance of the convolution product lies essentially in its regularization role. In this chapter, the authors introduce the convolution product in the space of distributions. They consider the convolution product of two integrable functions and then generalize to any two distributions by means of regular distributions. In order to introduce the convolution product of two distributions, it is most natural to define it for regular distributions, namely locally integrable functions. As for the convolution of a locally integrable function by a test function, the product of convolution of a distribution by a test function can be derived by obtaining the derivative of any of the two factors.