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448 CHAPTER 13 Fourier Series

Upon substituting these expressions for A 0 , A n , and B n into the Fourier series for F(x), we obtain
the conclusion of the theorem.

Valid term by term differentiation of a Fourier series requires stronger conditions.

THEOREM 13.5 Differentiation of Fourier Series

Let f be continuous on [−L, L] and suppose that f (−L) = f (L).Let f be piecewise

continuous on [−L, L]. Then the Fourier series of f on [−L, L] converges to f (x) on [−L, L]:
∞
1
f (x) = a 0 + (a n cos(nπx/L) + b n sin(nπx/L))
2
n=1
for −L ≤ x ≤ L. Further, at each x in (−L, L) at which f (x) exists, the term by term derivative

of the Fourier series converges to the derivative of the function:
∞
nπ
(−a n sin(nπx/L) + b n cos(nπx/L)).
f (x) =
L
n=1
The idea of a proof of this theorem is to begin with the Fourier series for f (x), noting

that this series converges to f (x) at each point where f exists. Use integration by parts to

relate the Fourier coefficients of f (x) to those for f (x), similar to the strategy used in proving

Theorem 13.4,
EXAMPLE 13.13
2
Let f (x) = x for −2 ≤ x ≤ 2. By the Fourier convergence theorem,
∞ n+1
4 16 (−1)
2
x = + cos(nπx/2)
3 π 2 n 2
n=1
2
for −2 ≤ x ≤ 2. Only cosine terms appear in this series because x is an even function. Now,

f (x) = 2x is continuous and f is twice differentiable for all x. Therefore, for −2 < x < 2,
∞ n+1
8 (−1)

f (x) = 2x = sin(nπx/2).
π n
n=1
This can be verified by expanding 2x in a Fourier series on [−2,2].
Fourier coefficients, and Fourier sine and cosine coefficients, satisfy an important set of
inequalities called Bessel’s inequalities.

THEOREM 13.6 Bessel’s Inequalities

L
Suppose g(x)dx exists.
0
1. The Fourier sine coefficients b n of g(x) on [0, L] satisfy
∞ L
2
2 2
b ≤ (g(x)) dx.
n
L 0
n=1

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October 14, 2010 14:57 THM/NEIL Page-448 27410_13_ch13_p425-464