Fourier series

Abstract

In the study of power series in Chapter 9, we saw that an analytic function f can be represented by a power series

$$f(x) = \sum\limits_{n = 0}^\infty {{c_n}{{(x - a)}^n}}$$

((10.1))

for all values of x within the radius of convergence of the series. We recall that f has derivatives of all orders and that the coefficients c _n in (10.1) are given by f⁽ⁿ⁾(a)/n!. In this chapter we shall be interested in series expansions of functions which may not be smooth. That is, we shall consider functions which may have only a finite number of derivatives at some points and which may be discontinuous at others. Of course, in such cases it is not possible to have expansions in powers of (x — a) such as (10.1). To obtain representations of unsmooth functions we turn to expansions in terms of trigonometric functions such as

$$1,{\kern 1pt} \cos x,\cos 2x,...,\cos nx,...$$

$${\kern 1pt} \sin x,\sin 2x,...,\sin nx,...$$