ABSTRACT
This chapter presents the topic of orthogonal expansions of functions with respect to a set of orthogonal functions. It introduces the fascinating example of such an orthogonal expansion—Fourier series. The chapter provides an essential part of the method of separation of variables, used to solve some linear partial differential equations. It analyzes the Jacques Charles Francois Sturm-Joseph Liouville problems and their applications. Many physical applications lead to linear differential equations that are subject to some auxiliary conditions that could be either boundary or boundedness conditions rather than initial conditions. To every such differential equation can be assigned a linear operator acting in some vector space of functions. People have enjoyed music since their appearance on the earth—it is one of the oldest pleasurable human activities. A musical tone is a steady periodic sound, which is often used in conjunction with pitch. The classical subject of Fourier series is about approximating periodic functions by sines and cosines.
