Page 459 - Advanced engineering mathematics

13.2 The Fourier Series of a Function 439

0.8

0.4

0
–3 –2 –1 0 1 2 3
x
–0.4

–0.8

FIGURE 13.12 The Gibbs phenomenon.

The Fourier series therefore converges to the function on (−π,π). This series is
∞
1
sin((2n − 1)x).
2n − 1
n=1
Figure 13.12 shows the fifth and twenty-fifth partial sums of this series, compared to the
function. Notice that both of these partial sums show a peak near 0, the point of discontinuity of
f . Since the partial sums S N approach the function as N →∞, we might expect these peaks to
flatten out, but they do not. Instead they remain roughly the same height, but move closer to the
y-axis as N increases. This is the Gibbs phenomenon.

Postscript We will add two comments on the ideas of this section, the first practical and the
second offering a broader perspective.

1. Writing and graphing partial sums of Fourier series are computation intensive activities.
Evaluating integrals for the coefficients is most efficiently done in MAPLE using the
int command, and partial sums are easily graphed using the sum command to enter the
partial sum and then the plot command for the graph.
2. Partial sums of Fourier series can be viewed from the perspective of orthogonal pro-
jections onto a subspace of a vector space (Sections 6.6 and 6.7). Let PC[−L, L] be
the vector space of functions that are piecewise continuous on [−L, L] and let S be the
subspace spanned by the functions
C 0 (x) = 1,C n (x) = cos(nπx/L) and S n (x) = sin(nπx/L) for n = 1,2,··· , N.

A dot product can be defined on PL[−L, L] by
L
f · g = f (x)g(x)dx.
−L
Using this dot product, these functions form an orthogonal basis for S.If f if piecewise
continuous on [−L, L], the orthogonal projection of f onto S is

N
f · C 0 f · C n f · S n
f S = C 0 + C n + S n .
C 0 · C 0 C n · C n S n · S n
n=1

Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

October 14, 2010 14:57 THM/NEIL Page-439 27410_13_ch13_p425-464