Fourier Series and Applications

Abstract

In a general course on functional analysis, one discusses a very general method of “Fourier analysis” in Hilbert space settings. Originally, the methods originated with the classical setting of real- or complex-valued periodic functions defined on the whole of \(\mathbb{R}\). In this chapter we focus our attention mainly on describing the elementary theory of classical Fourier series (with the help of specific kernels) which have become indispensable tools in the study of periodic phenomena in physics and engineering. These kernels are mainly used to prove the convergence of Fourier series, and the study of Fourier series has led to many important problems and theories in the mathematical sciences. As a result of the introduction of Fourier series, much of the development of modern mathematics has been influenced by the theory of trigonometric series. We ask a number of questions concerning the nature of Fourier series and provide answers to these questions.KeywordsFourier SeriesPeriodic FunctionFourier CoefficientTrigonometric SeriesFourier Series ExpansionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.