Environmental Engineering Reference
In-Depth Information
It is clear that the convolution satisfies
f
1
(
t
)
∗
f
2
(
t
)=
f
2
(
t
)
∗
f
1
(
t
)
,
f
1
(
t
)
∗
[
f
2
(
t
)+
f
3
(
t
)] =
f
1
(
t
)
∗
f
2
(
t
)+
f
1
(
t
)
∗
f
3
(
t
)
.
f
1
(
ω
)=
f
2
(
ω
)=
Convolution theorem.
If
F
[
f
1
(
t
)]
and
F
[
f
2
(
t
)]
,then
or
F
−
1
f
1
(
ω
)
f
2
(
ω
)
=
f
1
(
ω
)
f
2
(
ω
)
F
[
f
1
(
t
)
∗
f
2
(
t
)] =
f
1
(
t
)
∗
f
2
(
t
)
.
Proof
. By the definition of convolution,
+
∞
e
−
iω
t
d
t
F
[
f
1
(
t
)
∗
f
2
(
t
)] =
[
f
1
(
t
)
∗
f
2
(
t
)]
−
∞
+
∞
e
−
iω
t
d
t
+
∞
=
f
1
(
τ
)
f
2
(
t
−
τ
)
d
τ
−
∞
−
∞
+
∞
+
∞
f
1
(
ω
)
f
2
(
ω
)
,
e
−
iωτ
d
e
−
iω
(
t
−
τ
)
d
t
=
f
1
(
τ
)
τ
f
2
(
t
−
τ
)
=
−
∞
−
∞
where
+
∞
+
∞
f
2
e
−
iω
(
t
−
τ
)
d
t
e
−
iω
u
d
u
f
2
(
t
−
τ
)
=
f
2
(
u
)
=
(
ω
)
.
−
∞
−
∞
B.1.3 Generalized Functions and the
δ
-function
Generalized Functions Defined by the Functional
The functional is defined in function spaces to possess some good properties. The
commonly-used spaces are: (1) the
K-space
or
C
0
(
a
,
b
)
where functions are in-
finitely differentiable in
(
a
,
b
)
and vanished outside a finite interval, (2)
C
(
a
,
b
)
,(3)
L
2
where functions are continuous in
(
a
,
b
)
(
a
,
b
)
where functions are quadrati-
,and(4)
C
∞
(
cally integrable in
(
a
,
b
)
a
,
b
)
where functions have continuous deriva-
tives of any order up to infinity in
(
a
,
b
)
.Herethe
a
and the
b
can be
−
∞
and
+
∞
.
For example,
⎧
⎨
c
2
e
−
C
0
c
2
2
, |
x
| <
c
,
ϕ
(
x
)
∈
(
−
∞
,
+
∞
)
,
−|
x
|
ϕ
(
x
)=
(B.6)
⎩
0
,
|
x
|≥
c
.
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