## Abstract

Fourier Series is a way to represent a function \(f(t) \in L^2[a, b]\) of a continuous variable *t* with a countable number of coefficients \(c_k\). Oftentimes, however, we are interested in representing a finite number of data points \(\{f(t_k)\}_{k=0}^{N-1}\) which probably come as samples of a function of a continuous variable *t*. We may not have enough information to represent the original function, but we would still like to know what its Fourier Transform looks like. We have three primary goals for our discrete Fourier Analysis: (1) having an accurate representation, (2) being able to calculate the representation quickly and easily, and (3) knowing what the coefficients of that representation represent. We will now try to accomplish these goals.