Convolution Products

Abstract

This chapter introduces the convolution product for distributions and studies its basic properties. At first, we recall the convolution of functions and study the differentiability and integrability properties of the convolution product. It turns out that the test function space of strongly decreasing ${\mathcal C}^{\infty}$ function ${\mathcal{S}}(\mathbb{R}^n)$ is a commutative algebra when equipped with the convolution product. Then the convolution of a distribution with a test function is introduced. Through the concept of a regularizing sequence (of test functions) it is shown that every distribution is the limit of a sequence of ${\mathcal C}^{\infty}$ functions. The convolution product of two distributions is defined using the tensor product for distributions if these distributions satisfy the support condition. This convolution product is compatible with the convolution of functions and has similar differentiability properties. Dirac’s delta function is a unit for this convolution product. It turns out that the space of distributions on the real line with support in the positive half line $\mathcal{D}_+^{\prime}(\mathbb{R})$ is a commutative algebra under convolution, with unit and without divisors of zero.