Week 6 Lectures, Math 716, Tanveer
1
Fourier Series
In the context of separation of variable to find solutions of PDEs, we encountered
f
(
x
) =
∞
summationdisplay
n
=1
b
n
sin
nπx
l
for
x
∈
(0
, l
)
(1)
or
f
(
x
) =
a
0
2
+
∞
summationdisplay
n
=1
a
n
cos
nπx
l
for
x
∈
(0
, l
)
(2)
and in other cases
f
(
x
) =
a
0
2
+
∞
summationdisplay
n
=1
braceleftBig
a
n
cos
nπx
l
+
b
n
sin
nπx
l
bracerightBig
for
x
∈
(
−
l, l
)
(3)
The general representation (3) is called the Fourier representation of
f
(
x
) in the (
−
l, l
) interval,
while (1) and (2) are the Fourier sine and cosine representations of
f
(
x
) in the interval (0
, l
).
Our discussions revolve around some basic questions
1. When are represenations (1)(3) valid and in what sense?
2. How do we determine coefficients
a
n
,
b
n
in terms of
f
(
x
).
3. What conditions allow term by term differentiation of the FourierSeries.
2
General
L
2
theory
We need enough generality to be able to lay the framework for discussion of more general series
representations of
L
2
,
i.e.
square integrable functions than (1)(3), since they arise in other PDE
problems.
In the space
L
2
(
a, b
) of generally complex valued functions in the interval (
a, b
), we introduce
the
L
2
innerproduct:
(
f, g
) =
integraldisplay
b
a
f
(
x
)¯
g
(
x
)
dx
(4)
We note that the
L
2
norm is related through
bardbl
f
bardbl
= (
f, f
)
1
/
2
=
bracketleftBigg
integraldisplay
b
a

f

2
(
x
)
dx
bracketrightBigg
1
/
2
(5)
This may be generalized to any number of dimension, with
x
replaced by
x
and integration over
the interval (
a, b
), replaced by integration over appropriate
n
dimensional rectangle.
1
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Definition 1
A sequence
{
X
n
}
∞
n
=1
∈ L
2
(
a, b
)
is orthogonal if
(
X
n
, X
m
) = 0
iff
m
negationslash
=
n
(6)
This sequence is said to be orthonormal, if in addition
(
X
n
, X
n
) =
bardbl
X
n
bardbl
2
= 1
.
Theorem 2
Let
{
X
n
}
∞
n
=1
∈ L
2
(
a, b
)
be a orthogonal set of functions.
Let
bardbl
f
bardbl
<
∞
.
Let
N
be a fixed positive integer.
The choice of
A
n
that minimizes mean square error
E
N
=
bardbl
f
−
∑
N
n
=1
A
n
X
n
bardbl
2
is given by
A
n
=
(
f, X
n
)
bardbl
X
n
bardbl
2
(7)
Further, we have
bardbl
f
bardbl
2
≥
∞
summationdisplay
n
=1

(
f, X
n
)

2
bardbl
X
n
bardbl
2
Bessel inequality
Proof.
Define the square of the
E
N
=
bardbl
f
−
N
summationdisplay
n
=1
A
n
X
n
bardbl
2
=
parenleftBigg
f
−
N
summationdisplay
n
=1
A
n
X
n
, f
−
N
summationdisplay
n
=1
A
n
X
n
parenrightBigg
Expanding the above using properties of inner product and the orthogonality of
X
n
, we get
E
N
= (
f, f
)
−
N
summationdisplay
n
=1
A
n
(
X
n
, f
)
−
n
summationdisplay
n
=1
A
∗
n
(
f, X
n
) +
N
summationdisplay
n
=1
A
n
A
∗
n
(
X
n
, X
n
)
(8)
We minimize
E
n
as a function of 2
N
real variables,
{
(
c
n
, d
n
)
}
N
n
=1
, where
A
n
=
c
n
+
id
n
. On
taking partial derivatives, we get
∂E
N
∂c
n
=
−
(
X
n
, f
)
−
(
f, X
n
) + 2
c
n
(
X
n
, X
n
) = 0
implying
c
n
=
ℜ {
(
X
n
, f
)
}
bardbl
X
n
bardbl
2
Also,
∂E
N
∂d
n
=
−
i
(
X
n
, f
) +
i
(
f, X
n
) + 2
d
n
(
X
n
, X
n
) = 0
implying
d
n
=
ℑ {
(
X
n
, f
)
}
bardbl
X
n
bardbl
2
Therefore
A
n
=
c
n
+
id
n
is given by (7). Again with
A
n
given by (7), the corresponding
E
N
becomes
E
N
=
bardbl
f
bardbl
2
−
summationdisplay
n
≤
N

A
n

2
bardbl
X
n
bardbl
2
≥
0
Taking the limit of
N
→ ∞
, we obtain Bessel inequality.
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 Spring '08
 Tanveer,S
 Fourier Series, Sin, Continuous function, Uniform convergence

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