Applications of Commutative Harmonic Analysis

In this set of notes we shall investigate applications of commutative harmonic analysis, which for us will mean the Fourier transform on commutative groups. The focus will be on examples, and these will come from a range of settings including functional analysis, number theory, and probability. Most of what we do will focus on quantitative estimates related to finite groups, and in light of this there is no one specific reference to be recommended. There are a number which give a flavour of different aspects of what we are interested in, and we shall try give references as we proceed. Before we introduce the Fourier transform it is worth providing some motivation for its definition. For us this motivation will come from convolution. Convolution is something which will have come up in a variety of settings and is an exceptionally useful mathematical tool. We shall begin with some examples and then extend it to the more general setting in which we are interested.