Dirichlet conditions

It is important to establish simple criteria which determine when a Fourier series converges. In this section we will develop conditions on f(x) that enable us to determine the sum of the Fourier series. One quite useful method to analyse the convergence properties is to express the partial sums of a Fourier series as integrals. Riemann and Fejer have since provided other ways of summing Fourier series. In this section we limit the study of convergence to functions that are piecewise smooth on a given interval.

Definition 5 (Piecewise smooth function)
A function f is piecewise smooth on an interval if both f and f' are piecewise continuous on the interval.

Theorem 2
Suppose that f is piecewise smooth and periodic.
Then the series (1) with coefficients (2) converges to

1.

f(x) if x is a point of continuity.

2.

if x is a point of discontinuity.

This means that, at each x between -L and L, the Fourier series converges to the average of the left and the right limits of f(x) at x. If fis continuous at x, then the left and the right limits are both equal to f(x), and the Fourier series converges to f(x) itself. If f has a jump discontinuity at xthen the Fourier series converges to the point midway in the gap at this point.

Remark 3
Let f be a given piecewise continuous function. We say that f is standardised if its values at points x_i of discontinuity are given by

Remark 4
The conditions imposed on f(x) are sufficient but not necessary, i.e if the conditions are satisfied the convergence is guaranteed. However, if they are not satisfied the series may or may not converge.

Theorem 3 (Bessel's inequality)
Suppose that f is integrable on the interval [0,T]. Let a_n, b_n, c_n be the Fourier coefficients of f. Then

Theorem 4 (Riemann lemma)
Let f be integrable and a_n and b_n be the Fourier coefficients of f. Then

which means

Theorem 5 (Parseval's identity)

if a_n and b_n are the Fourier coefficients corresponding to f(x) and if f(x)satisfies the Dirichlet conditions.