Math 602
Homework 7
The problems here are well known results.
You need to show all your
work and explanations.
1.
(i) Check that the functions 1
,
sin(
nx
)
,
cos(
nx
)
,
(
n
= 1
,
2
,
3
. . .
) form
an orthogonal system in
L
2
[

π, π
].
(ii) Normalize them to obtain an orthonormal system.
(iii) Assuming that
f
∈
L
2
[

π, π
] has its Fourier series expansion
f
=
a
0
2
+
∞
X
n
=1
[
a
n
cos(
nx
) +
b
n
sin(
nx
)]
(1)
verify that the Fourier coefficients
a
n
and
b
n
are given by the formulas
a
n
=
1
π
Z
π

π
f
(
x
) cos(
nx
)
dx ,
b
n
=
1
π
Z
π

π
f
(
x
) sin(
nx
)
dx,
2.
(i) Check that the functions
e
inx
, n
∈
Z
form an orthogonal system
in the complex valued functions
L
2
[

π, π
].
(ii) Normalize them to obtain an orthonormal system.
(iii) Assuming that
f
∈
L
2
[

π, π
] has its Fourier series
f
=
∞
X
n
=
∞
ˆ
f
n
e
inx
(2)
verify that the Fourier coefficients are given by
ˆ
f
n
=
1
2
π
Z
π

π
f
(
x
)
e

inx
dx
(iv) Verify that the Fourier coefficients
a
n
, b
n
,
ˆ
f
n
of a function
f
(
x
) are
related by the formulas
a
n
=
ˆ
f
n
+
ˆ
f

n
for
n
= 0
,
1
,
2
, . . . ,
b
n
=
i
(
ˆ
f
n

ˆ
f

n
) for
n
= 1
,
2
, . . .
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 Spring '12
 COSTIN
 Math, Fourier Series, Orthonormal basis, Fourier coefficients

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