2.1 Discrete Fourier Transform

Fourier analysis started even before Fourier, but in the context of Fourier’s work on the Analytic Theory of Heat, it began with a claim regarding expansions of functions as trigonometrical series, what we now call Fourier series. Such expansions form the framework for writing down solutions to the heat equation in terms of initial temperature distributions. [refer to Enrique A. Gonzalez-Velasco The American Mathematical Monthly, Vol. 99, No. 5 (May, 1992), pp. 427-441 doi:10.2307/2325087] We will not follow the historic development of Fourier series. Instead, we will begin with a finite version of the Fourier transform, the so-called discrete Fourier transform, then proceed to discuss Fourier series expansions of periodic functions and Fourier transforms of functions defined on the whole real line as limiting cases. As a note regarding terminology, the word discrete when applied to Fourier transforms sometimes applies to the finite case, but it also sometimes applies to the case of sequences defined on the integers. Here we will always use the term discrete in reference to the finite case only.