VII - Fourier Series

Abstract

This chapter focuses on “function space” as opposed to a three-dimensional “vector space.” This function space is infinite dimensional, in the sense that an infinite sequence of mutually orthogonal functions is needed to represent an arbitrary function. Infinite-dimensional space is more complicated than the three-dimensional space. No sequence of mutually orthogonal functions is satisfactory. Some restrictions must be placed on the class of functions that are to be represented by a series of the orthogonal functions—that is, by a Fourier series. The chapter lists formulas for the Fourier coefficients of an arbitrary function corresponding to some specific sets of orthogonal functions. Conditions under which the series converge to the function is discussed. There is no general procedure for determining whether an arbitrary orthogonal set is complete. However, completeness has been established for certain important classes of orthogonal sets. One of these is the class of orthogonal polynomials on a bounded interval. Another consists of the sets of eigenfunctions of self-adjoint eigenvalue problems.