Can you hear the shape of a drum? How does the FBI store 200 million fingerprints efficiently? What is behind the JPEG standard used for picture-compression on the web? Here we will study how to answer such questions by decomposing a function into "special functions" such as sines and cosines (known as Fourier Series), or the more modern analogue known as wavelets. For example, a musical tone can be broken up into its overtones using Fourier analysis, making it easier to analyze. Similarly, a visual image can be broken up into its wavelet components. We will discuss Fourier series, continuous and discrete wavelet transforms, the Heisenberg uncertainty principle, and possibly frames and MRA wavelets. Evaluation will be based primarily on homeworks and exams. Prerequisites: Mathematics 301 or Mathematics 305 and Mathematics 211.