Abstract: Fourier analysis is a powerful tool for analyzing certain kinds of data. It has broad applications to many fields, including electrical engineering and signal and image processing. For example, Fourier analysis lies at the heart of the JPEG compression algorithm and many common operations on audio signals. These all rely on the Fast Fourier Transform, which is generally considered to be one of the most important algorithms of the twentieth century. In principle, Fourier analysis involves breaking a signal down into individual frequencies. This corresponds to writing a function in terms of  trigonometric functions---such a representation is called a Fourier series. In addition to some interesting applications, we will discuss
procedures for constructing a Fourier series from a given function or data set. This can be done using techniques from calculus, or with  software such as Mathematica. There are deeper mathematical ideas at play as well, and we will hint at these along the way.