2 - Fourier Series

Abstract

This chapter gives sufficient conditions for the convergence of trigonometric series and presents some related results. Fourier coefficients are defined by integrals and the chapter gives a brief examination of the concept of integral is in order along with the discussion on the convergence of Fourier series. Moreover, once the concept of integral is established on a firm foundation, the chapter presents a result—obtained later by a former student of Dirichlet—that allows giving a shorter proof of a sharper convergence theorem than the original one by Dirichlet. This result, the so-called Riemann–Lebesgue theorem, is a very powerful tool. The popular concept of integral in the eighteenth century was that of prederivative or indefinite integral. It was sufficient for all their applications and easy to explain. In a practical situation a Fourier series cannot be used in its entirety to approximate a given function since one must necessarily take only a finite number of terms. This raises the question of how well a partial sum of a Fourier series approximates the function to which it converges.