Environmental Engineering Reference
In-Depth Information
Definition 1.
Suppose that there exists a real value corresponding to every function
y
according to a certain rule; such a corresponding
relation is called the
functional
and is denoted by
F
(
x
)
in the function space
{
y
(
x
)
}
.
If
F
represents a functional from
K
-space to
R
1
and satisfies lim
n
=
F
(
y
(
x
))
F
(
ϕ
n
)=
F
(
ϕ
)
,
→
∞
∀{
ϕ
n
}⊂
K
when lim
n
→
∞
ϕ
n
=
ϕ
, in particular, then
F
is said to be continuous at point
ϕ
.If
F
is continuous at all points in
K
, it is called a
continuous functional
in
K
. If,
for all
R
1
,
ϕ
1
,
ϕ
2
∈
K
and
k
1
,
k
2
∈
F
(
k
1
ϕ
1
+
k
2
ϕ
2
)=
k
1
F
(
ϕ
1
)+
k
2
F
(
ϕ
2
)
,
the
F
is called a
linear functional
.
Definition 2.
A linear continuous functional in
K
is called a
generalized function in
K
and is denoted by
F
(
ϕ
)
or
(
F
,
ϕ
)
,
ϕ
∈
K
.
Remark.
Generalized functions depend on function spaces and differ from classi-
cal functions. Once its value corresponding to every element in a function space is
known, it is regarded as fixed. In a specified function space, we may express the
generalized functions in various formats. The commonly used one is the integral of
an inner product. If
f
is integrable in
R
1
,
(
x
)
+
∞
F
f
(
ϕ
)=
f
(
x
)
ϕ
(
x
)
d
x
,
∀
ϕ
(
x
)
∈
K
(B.7)
−
∞
is a generalized function in
K
and is often denoted by
(
f
,
ϕ
)
or
f
,
ϕ
,
∀
ϕ
(
x
)
∈
K
.
Generalized Functions Defined by Equivalent Classes of Basic Sequences
C
0
(
Definition 3.
Suppose that the function sequences
{
f
n
(
x
)
}⊂
a
,
b
)
.
b
If lim
n
→
∞
f
n
(
x
)
ϕ
(
x
)
d
x
exists
∀
ϕ
∈
K
,the
{
f
n
}
is called a
basic sequence of K
.If
a
two basic sequences
{
f
n
}
and
{
g
n
}
of
K
satisfy
b
b
lim
n
f
n
(
x
)
ϕ
(
x
)
d
x
=
lim
n
g
n
(
x
)
ϕ
(
x
)
d
x
,
→
∞
→
∞
a
a
the
{
f
n
}
and the
{
g
n
}
are called
equivalent
and denoted by
{
f
n
}∼{
g
n
} ,
K
.
Definition 4.
The equivalent classes of basic sequences of
K
-space are called
gener-
alized functions
and denoted by
f
=
{
f
n
}
or
f
n
→
f
.Then
{
f
n
}
is said to be
weakly
convergent
to
f
. Also, denote
b
b
,
ϕ
=
(
)
ϕ
(
)
=
(
)
ϕ
(
)
,
∀
ϕ
∈
.
f
f
x
x
d
x
lim
n
f
n
x
x
d
x
K
(B.8)
→
∞
a
a
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