Page 469 - Advanced engineering mathematics

13.4 Integration and Differentiation of Fourier Series 449

2. The Fourier cosine coefficients a n of g(x) on [0, L] satisfy
∞ L
1 2 2 2 2
a + a ≤ (g(x)) dx.
0
n
2 L 0
n=1
L
3. If g(x)dx exists, then the Fourier coefficients of f (x) on [−L, L] satisfy
−L
∞ L
1 2 2 2 1 2
a + (a + b ) ≤ (g(x)) dx.
n
0
n
2 L −L
n=1
In particular, the sum of the squares of the coefficients in a Fourier series (or cosine or sine
series) converges.
We will prove conclusion (1). The argument is notationally simpler than that for conclusions
(2) and (3), but contains the ideas involved.
Proof of (1) The Fourier sine series of g(x) on [0, L] is
∞

nπx
b n sin ,
L
n=1
where
2 L nπx
b n = g(x)sin dx.
L 0 L
The Nth partial sum of this sine series is
N

nπx
S N (x) = b n sin .
L
n=1
Then
L
2
0 ≤ (g(x) − S N (x)) dx
0
L L L
2
2
= (g(x)) dx − 2 g(x)S N (x)dx + (S N (x)) dx
0 0 0

L L
N
nπx
2
= (g(x)) dx − 2 g(x) b n sin dx
L
0 0 n=1

N N

nπx kπx
L
+ b n sin b k sin dx
L L
0
n=1 k=1
L N L

nπx
2
= (g(x)) dx − 2 b n g(x)sin dx
0 n=1 0 L
N N L
nπx kπx
+ b n b k sin sin dx
L L
n=1 k=1 0
L N N
L
2
= (g(x)) dx − b n (Lb n ) + b n b n .
2
0
n=1 n=1
Here we have used the fact that

L
nπx kπx 0 for n = k,
sin sin dx =
0 L L L/2for n = k.
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October 14, 2010 14:57 THM/NEIL Page-449 27410_13_ch13_p425-464