FOURIER ANALYSIS USING REPRESENTATION THEORY

Abstract. In this paper, it is our goal to develop the concept of Fourier transforms by using Representation Theory. We begin by laying basic def-initions that will help us understand the definition of a representation of a group. Then, we define representations and provide useful definitions and the-orems. We study representations and key theorems about representations such as Schur’s Lemma. In addition, we develop the notions of irreducible repre-senations, *-representations, and equivalence classes of representations. After doing so, we develop the concept of matrix realizations of irreducible represen-ations. This concept will help us come up with a few theorems that lead up to our study of the Fourier transform. We will develop the definition of a Fourie