Fourier Sine Series Examples
16th November 2007
The Fourier sine series for a function
f
(
x
)
defined on
x
∈
[0
,
1]
writes
f
(
x
)
as
f
(
x
) =
∞
n
=1
b
n
sin(
nπx
)
for some coefficients
b
n
. Because of orthogonality, we can compute the
b
n
very simply:
for any given
m
, we integrate both sides against
sin(
mπx
). In the summation, this gives
zero for
n
=
m
, and
1
0
sin
2
(
mπx
) = 1
/
2
for
n
=
m
, resulting in the equation
b
m
= 2
1
0
f
(
x
) sin(
mπx
)
dx.
Fourier claimed (without proof) in 1822 that
any
function
f
(
x
)
can be expanded in
terms of sines in this way, even discontinuous function. This turned out to be false for
various badly behaved
f
(
x
)
, and controversy over the exact conditions for convergence
of the Fourier series lasted for well over a century, until the question was finally settled
by Carleson (1966) and Hunt (1968): any function
f
(
x
)
where

f
(
x
)

p
dx
is finite
for some
p >
1
has a Fourier series that converges
almost everywhere
to
f
(
x
)
[except
at isolated points]. At points where
f
(
x
)
has a jump discontinuity, the Fourier series
converges to the midpoint of the jump. So, as long as one does not care about crazy di
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 Spring '08
 sc
 Fourier Series, Continuous function, Fourier sine series

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