Fourier Analysis Using Representation Theory

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