KIAM HEONG KWA
The Gibbs phenomenon, named after the American physicist Josiah
Willard Gibbs, is the peculiar manner in which the Fourier series of
a piecewise continuously diﬀerentiable periodic function behaves at a
jump discontinuity: the partial sums of the Fourier series have large
oscillations near the jump, which might increase the maxima of the
partial sums above that of the function itself. The overshoot does not
die out as the frequency increases, but approaches a ﬁnite limit.
Following [1], we illustrate the Gibbs phenomenon with the
π
periodic
function
f
(
x
) such that
(1)
f
(
x
) =
1
2
for 0
< x < π,

1
2
for

π < x <
0
.
It Fourier series is
F
[
f
](
x
) =
a
0
2
+
∞
X
n
=1
(
a
n
cos
nx
+
b
n
sin
nx
)
(2)
=
∞
X
odd
n
=1
2 sin
nx
nπ
and the corresponding partial sums are
F
N
[
f
](
x
) =
a
0
2
+
N
X
n
=1
(
a
n
cos
nx
+
b
n
sin
nx
)
(3)
=
N
X
odd
n
=1
2 sin
nx
nπ
Date
: February 12, 2011.
1
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 Winter '11
 Kwa
 Differential Equations, Equations, Partial Differential Equations, Fourier Series, Sin, Willard Gibbs

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