Environmental Engineering Reference
In-Depth Information
Similarly,
+
∞
e
iω
x
ω
=
0
1
2
1
2
F
−
1
e
iω
x
d
[
δ
(
ω
)] =
δ
(
ω
)
ω
=
π
π
−
∞
1
2
=
π
.
Thus
F
−
1
[
πδ
(
ω
)] =
[
]=
πδ
(
ω
)
.
2
1or
F
1
2
The Generalized Fourier Transformation and its Properties
Suppose that there exist weak limits of the function sequences
{
u
n
(
x
)
}
and
{
F
[
u
n
(
x
)]
}
such that
u
n
(
x
)
→
u
(
x
)
and
F
[
u
n
(
x
)]
→
u
(
ω
)
.
The
u
(
ω
)
is called the
generalized Fourier transformation
of
u
(
x
)
. As before, we
denote this as
or
F
−
1
F
[
u
(
x
)] =
u
(
ω
)
[
u
(
ω
)] =
u
(
x
)
.
By using the notation of Fourier transformations, we have
+
∞
e
−
iω
x
d
x
F
[
u
(
x
)] =
u
(
x
)
,
(B.19)
−
∞
+
∞
1
2
F
−
1
e
iω
x
d
[
u
(
ω
)] =
u
(
ω
)
ω
.
(B.20)
π
−
∞
Note that the integrals in Eq. (B.20) are not the integrations of the classical Fourier
transformation.
It can be shown that the generalized Fourier transformation shares fundamental
properties with the classical Fourier transformation except that the integral property
now reads
F
x
−
∞
f
(
ω
)
i
f
f
(
ξ
)
d
ξ
=
+
π
(
0
)
δ
(
ω
)
.
ω
Example 3.
Find the generalized Fourier transformation of the Heaviside function
0
,
<
,
x
0
H
(
x
)=
,
<
.
1
0
x
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