Environmental Engineering Reference
In-Depth Information
Taking a Fourier transformation with respect to
y
leads to
+
∞
+
∞
f
e
−
i
(
ω
1
x
+
ω
2
y
)
d
x
d
y
(
ω
1
,
ω
2
,
z
)=
f
(
x
,
y
,
z
)
,
−
∞
−
∞
+
∞
+
∞
1
f
e
i
(
ω
1
x
+
ω
2
y
)
d
f
(
x
,
y
,
z
)=
(
ω
1
,
ω
2
,
z
)
ω
1
d
ω
2
.
(
2
π
)
2
−
∞
−
∞
Finally, a Fourier transformation with respect to
z
yields
+
∞
+
∞
+
∞
f
e
−
i
(
ω
1
x
+
ω
2
y
+
ω
3
z
)
d
x
d
y
d
z
(
ω
1
,
ω
2
,
ω
3
)=
f
(
x
,
y
,
z
)
,
(B.21)
−
∞
−
∞
−
∞
+
∞
+
∞
+
∞
1
f
e
i
(
ω
1
x
+
ω
2
y
+
ω
3
z
)
d
f
(
x
,
y
,
z
)=
(
ω
1
,
ω
2
,
ω
3
)
ω
1
d
ω
2
d
ω
3
.
(B.22)
(
2
π
)
3
−
∞
−
∞
−
∞
The former is called the
triple Fourier transformation
of
f
(
x
,
y
,
z
)
. The latter is called
the
inverse triple Fourier transformation
.The
f
(
ω
1
,
ω
2
,
ω
3
)
is called the
image
is the
inverse image function
of
f
function
of
f
(
x
,
y
,
z
)
,and
f
(
x
,
y
,
z
)
(
ω
1
,
ω
2
,
ω
3
)
.De-
note the points
(
x
,
y
,
z
)
and
(
ω
1
,
ω
2
,
ω
3
)
as
r
=
x
i
+
y
j
+
z
k
and
ω
=
ω
1
i
+
ω
2
j
+
ω
3
k
,
respectively. Equations (B.21) and (B.22) can thus be written as
+
∞
+
∞
+
∞
f
e
iω
·
r
d
x
d
y
d
z
(
ω
)=
f
(
r
)
,
−
∞
−
∞
−
∞
+
∞
+
∞
+
∞
1
f
e
iω
·
r
d
f
(
r
)=
(
ω
)
ω
1
d
ω
2
d
ω
3
,
(
2
π
)
3
−
∞
−
∞
−
∞
which are denoted as
F
−
1
f
(
ω
)
.
f
(
ω
)=
F
[
f
(
r
)]
,
f
(
r
)=
Remark.
By following a similar approach, we can also define other multiple Fourier
transformations such as the double Fourier transformation. The multiple transfor-
mation also shares the same properties as the Fourier transformation. For a function
f
(
x
,
y
,
z
)
of three variables, for example, we have
F
∂
F
∂
F
∂
f
f
f
=
i
ω
1
F
[
f
]
,
=
i
ω
2
F
[
f
]
,
=
i
ω
3
F
[
f
]
.
∂
x
∂
y
∂
z
For two functions
f
1
(
x
,
y
,
z
)
and
f
2
(
x
,
y
,
z
)
of three variables
x
,
y
and
z
, their convo-
lution is defined by
f
1
(
x
,
y
,
z
)
∗
f
2
(
x
,
y
,
z
)
+
∞
+
∞
+
∞
=
f
1
(
t
1
,
t
2
,
t
3
)
f
2
(
x
−
t
1
,
y
−
t
2
,
z
−
t
3
)
d
t
1
d
t
2
d
t
3
.
−
∞
−
∞
−
∞
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