This preview shows pages 1–3. Sign up to view the full content.
ANALYSIS TOOLS WITH APPLICATIONS
197
11.
Approximation Theorems and Convolutions
Let
(
X,
M
,µ
)
be a measure space,
A
⊂
M
an algebra.
Notation 11.1.
Let
S
f
(
A
)
denote those simple functions
φ
:
X
→
C
such that
φ
−
1
(
{
λ
}
)
∈
A
for all
λ
∈
C
and
µ
(
φ
6
=0)
<
∞
.
For
φ
∈
S
f
(
A
)
and
p
∈
[1
,
∞
)
,

φ

p
=
P
z
6
=0

z

p
1
{
φ
=
z
}
and hence
Z

φ

p
dµ
=
X
z
6
=0

z

p
µ
(
φ
=
z
)
<
∞
so that
S
f
(
A
)
⊂
L
p
(
µ
)
.
Lemma 11.2
(Simple Functions are Dense)
.
The simple functions,
S
f
(
M
)
,
form
adensesubspaceo
f
L
p
(
µ
)
for all
1
≤
p<
∞
.
Proof.
Let
{
φ
n
}
∞
n
=1
be the simple functions in the approximation Theorem
7.12. Since

φ
n

≤

f

for all
n, φ
n
∈
S
f
(
M
)
(verify!) and

f
−
φ
n

p
≤
(

f

+

φ
n

)
p
≤
2
p

f

p
∈
L
1
.
Therefore, by the dominated convergence theorem,
lim
n
→∞
Z

f
−
φ
n

p
dµ
=
Z
lim
n
→∞

f
−
φ
n

p
dµ
=0
.
Theorem 11.3
(Separable Algebras implies Separability of
L
p
—Spaces)
.
Suppose
1
≤
∞
and
A
⊂
M
is an algebra such that
σ
(
A
)=
M
and
µ
is
σ

f
nite on
A
.
Then
S
f
(
A
)
is dense in
L
p
(
µ
)
.
Moreover, if
A
is countable, then
L
p
(
µ
)
is
separable and
D
=
{
X
a
j
1
A
j
:
a
j
∈
Q
+
i
Q
,A
j
∈
A
with
µ
(
A
j
)
<
∞
}
is a countable dense subset.
Proof. First Proof.
Let
X
k
∈
A
be sets such that
µ
(
X
k
)
<
∞
and
X
k
↑
X
as
k
→∞
.
k
∈
N
let
H
k
denote those bounded
M
— measurable functions,
f,
on
X
such that
1
X
k
f
∈
S
f
(
A
)
L
p
(
µ
)
.
It is easily seen that
H
k
is a vector space closed
under bounded convergence and this subspace contains
1
A
for all
A
∈
A
.
Therefore
by Theorem 8.12,
H
k
is the set of all bounded
M
— measurable functions on
X.
f
∈
L
p
(
µ
)
,
the dominated convergence theorem implies
1
X
k
∩
{
f

≤
k
}
f
→
f
in
L
p
(
µ
)
as
k
.
We have just proved
1
X
k
∩
{
f

≤
k
}
f
∈
S
f
(
A
)
L
p
(
µ
)
for all
k
and hence it follows that
f
∈
S
f
(
A
)
L
p
(
µ
)
.
The last assertion of the theorem is
a consequence of the easily veri
f
ed fact that
D
is dense in
S
f
(
A
)
relative to the
L
p
(
µ
)
—norm
.
Second Proof.
Given
±>
0
,
by Corollary 8.42, for all
E
∈
M
such that
µ
(
E
)
<
∞
,
there exists
A
∈
A
such that
µ
(
E
4
A
)
<±.
Therefore
(11.1)
Z

1
E
−
1
A

p
dµ
=
µ
(
E
4
A
)
<±
.
This equation shows that any simple function in
S
f
(
M
)
may be approximated
arbitrary well by an element from
D
and hence
D
is also dense in
L
p
(
µ
)
.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document198
BRUCE K. DRIVER
†
Corollary 11.4
(Riemann Lebesgue Lemma)
.
Suppose that
f
∈
L
1
(
R
,m
)
,
then
lim
λ
→
±
∞
Z
R
f
(
x
)
e
iλx
dm
(
x
)=0
.
Proof.
Let
A
denote the algebra on
R
generated by the half open intervals, i.e.
A
consists of sets of the form
n
a
k
=1
(
a
k
,b
k
]
∩
R
where
a
k
k
∈
¯
R
.
By Theorem 11.3given
±>
0
there exists
φ
=
P
n
k
=1
c
k
1
(
a
k
,b
k
]
with
a
k
k
∈
R
such that
Z
R

f
−
φ

dm < ±.
Notice that
Z
R
φ
(
x
)
e
iλx
dm
(
x
)=
Z
R
n
X
k
=1
c
k
1
(
a
k
,b
k
]
(
x
)
e
iλx
dm
(
x
)
=
n
X
k
=1
c
k
Z
b
k
a
k
e
iλx
dm
(
x
n
X
k
=1
c
k
λ
−
1
e
iλx

b
k
a
k
=
λ
−
1
n
X
k
=1
c
k
¡
e
iλb
k
−
e
iλa
k
¢
→
0
as

λ

→∞
.
This is the end of the preview. Sign up
to
access the rest of the document.
 Three '10
 Smith
 Algebra, Approximation, The Land

Click to edit the document details