ABSTRACT
Trigonometric and Fourier series constitute one of the oldest parts of analysis. They play a central role in the study of sound, heat conduction, electromagnetic waves, mechanical vibrations, signal processing, and image analysis and compression. This chapter is concerned with three main questions: given a function f, how to calculate the coefficients an, bn; once the series for f has been calculated, can we determine that it converges, and that it converges to f; and how can we use Fourier series to solve a differential equation. The chapter looks at some classical calculations that were first performed by Leonhard Euler. The study of convergence of Fourier series is both deep and subtle. The chapter very briefly describes a few of the basic results. One of the great ideas of twentieth-century mathematics is that many other spaces—sometimes abstract spaces, and sometimes infinite-dimensional spaces—can be equipped with an inner product that endows that space with a useful geometry.
