Environmental Engineering Reference
In-Depth Information
holds for any integer
n
≥
0and
k
≥
0, then
u
(
x
)
is called a
fast-decreasing function
.
0
,
t
<
0
,
An example is the exponentially-decreasing function
f
(
t
)=
e
−
β
t
,
0
≤
t
,
β
>
0
that we discussed before. The
n
π
e
−
nx
2
(
n
=
1
,
2
, ···
)
is a sequence of fast-
decreasing functions.
By Definition 4, for all
C
0
(
−
∞
,
+
∞
)
ϕ
(
x
)
∈
we have
+
∞
+
∞
lim
n
→
∞
u
n
(
x
)
ϕ
(
x
)
d
x
=
u
(
x
)
ϕ
(
x
)
d
x
.
(B.12)
−
∞
−
∞
Similarly,
+
∞
+
∞
lim
ε
→
+
u
ε
(
x
)
ϕ
(
x
)
d
x
=
u
(
x
)
ϕ
(
x
)
d
x
.
(B.13)
0
−
∞
−
∞
We call the
u
(
x
)
the weak limit of
{
u
n
(
x
)
}
or
{
u
ε
(
x
)
}
, and denote
weak
=
weak
=
lim
n
u
n
(
x
)
u
(
x
)
or
lim
ε
→
+
0
u
ε
(
x
)
u
(
x
)
.
→
∞
Equations (B.12) and (B.13) show that we can actually interchange the order of lim-
its and integration. We regard the above-mentioned functions, which cannot undergo
the Fourier transformation, as the weak limits of some fast-decreasing functions. It
can easily be shown that a classical limit must also be a weak limit, but a weak limit
is not necessarily a classical limit.
We can use the weak limits of fast-decreasing functions to extend the definition
of the Fourier transformation.
Examples of Generalized Fourier Transformation
Example 1
. Find the image function of unit function
0
,
x
<
0
,
I
(
x
)=
1
,
x
≥
0
.
Solution
. The Fourier transformation
I
(
x
)
does not exist under the classical defini-
tion. Consider the product of
I
(
x
)
and an exponentially-decreasing function
f
(
x
)
;
we have
e
−
β
x
lim
β
→
0
I
(
x
)
f
(
x
)=
lim
β
→
0
I
(
x
)
=
I
(
x
)
.
(B.14)
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