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7.5 Approximation by polynomials
121
Convolution is a smoothing operation.
Theorem 7.5.5.
For
f,g
∈ R
(
D
)
,
f
*
g
is continuous,
D
compact.
Proof.
First, prove it for the case when
f,g
are continuous. Fix
ε >
0 and
c
∈
D
. Compact
support gives a bound for the continuous function
f
, say

f
(
x
)

< B
. Also, it gives uniform
continuity of
g
, so ﬁnd
δ >
0 such that

x

c

< δ
=
⇒ 
(
x

y
)

(
c

y
)

< δ
=
⇒ 
g
(
x

y
)

g
(
c

y
)

< ε.
Now for

x

c

< δ
, we have

(
f
*
g
)(
x
)

(
f
*
g
)(
c
)

=
ﬂ
ﬂ
ﬂ
ﬂ
Z
D
f
(
y
)
g
(
x

y
)

g
(
c

y
)
dy
ﬂ
ﬂ
ﬂ
ﬂ
≤
Z
D

f
(
y
)
 · 
g
(
x

y
)

g
(
c

y
)

dy
≤
Bε
· 
spt
f

.
Now suppose that
f,g
are not necessarily continuous. Let
{
f
n
} ⊆
C
(
D
) be a sequence
with

f
n
(
x
)
 ≤
B
and
R
D

f
n
(
x
)

f
(
x
)

dx
n
→∞
→
0, and similarly
{
g
n
} ⊆
C
(
D
). Then
f
*
g

f
n
*
g
n
= (
f

f
n
)
*
g
+
f
n
*
(
g

g
n
)
.
Consequently, we have
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 Fall '11
 Wong
 Polynomials, Approximation

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