In this chapter, we examine the trigonometric expansion of a function f(x) defined on an interval such as –π ≤ x ≤ π. A trigonometric expansion is a sum of the form
where the sum could be finite or infinite. Why should we care about expressing a function in such a way? As the following sections show, the answer varies depending on the application we have in mind.
1.1.1 Historical Perspective
Trigonometric expansions arose in the 1700s, in connection with the study of vibrating strings and other, similar physical phenomena; they became part of a controversy over what constituted a general solution to such problems, but they were not investigated in any systematic way. In 1808, Fourier wrote the first version of his celebrated memoir on the theory of heat, Théorie Analytique de la Chaleur, which was not published until 1822. In it, he made a detailed study of trigonometric series, which he used to solve a variety of heat conduction problems.
Fourier’s work was controversial at the time, partly because he did make unsubstantiated claims and overstated the scope of his results. In addition, his point of view was new and strange to mathematicians of the day. For instance, in the early 1800s a function was considered to be any expression involving known terms, such as powers of x, exponential functions, and trigonometric functions. The more abstract ...