This video will show how to approximate a function with a Fourier series, which is an infinite sum of sines and cosines. We will discuss how these sines and cosines form a basis for the space of functions. Book Website: http://databookuw.com/ Book PDF: http://databookuw.com/databook.pdf These lectures follow Chapter 2 from: "Data-Driven Science and Engineering: Machine Learning, Dynamical Systems, and Control" by Brunton and Kutz Amazon: https://www.amazon.com/Data-Driven-Science-Engineering-Learning-Dynamical/dp/1108422098/ Brunton Website: eigensteve.com