Convolution

The aim of this chapter is to present a systematic study of the convolution. We will actually introduce two very useful tools in practice for studying the local averages of a function: the convolution product \( \textit{f} \ast \textit{g} \) and the maximal function M (f). These two tools will allow us to study the approximations of the identity introduced by Friedrichs and to prove fundamental inequalities due to Hardy, Littlewood and Sobolev. In the last two sections,we will study several results related to the concept of convolution and which were at the heart of the concerns of harmonic analysis in the twentieth century: the Calderón’s reconstruction formula, the wavelet decomposition of Grossmann and Morlet as well as Wiener’s Tauberian theorem.