38
1 The Fourier Transform on
L
1
(
R
)
which are dense subspaces of
L
p
(
R
). On these domains,
P
:
D
P
→
L
p
(
R
) and
M
:
D
M
→
L
p
(
R
). Show, however, that
P
and
M
are unbounded even when
restricted to these domains, i.e.,
sup
f
∈
D
P
,
bardbl
f
bardbl
p
=1
bardbl
Pf
bardbl
p
=
∞
=
sup
f
∈
D
M
,
bardbl
f
bardbl
p
=1
bardbl
Mf
bardbl
p
.
1.5 Approximate Identities
Although
L
1
(
R
) is closed under convolution, we have seen that it has no
identity element. In this section we will show that there are functions in
L
1
(
R
)
that are “almost” identity elements for convolution. We will construct families
of functions
{
k
λ
}
λ>
0
which have the property that
f
∗
k
λ
converges to
f
in
L
1
(
R
) (and in fact in many other senses, depending on what space
f
belongs
to). If
k
λ
is a “nice” function, then
f
∗
k
λ
will inherit that niceness, and so we
will have a nice function that is arbitrarily close to
f
. Using this procedure we
will be able to show that many spaces of nice functions are dense in
L
1
(
R
),
including
C
c
(
R
),
C
m
c
(
R
) for each
m
∈
N
,
and even
C
∞
c
(
R
).
1.5.1 Definition and Existence of Approximate Identities
The properties that a family
{
k
λ
}
λ>
0
will need to possess in order to be an
approximate identity for convolution are listed in the next definition.
Definition 1.51.
An
approximate identity
or a
summability kernel
is a family
{
k
λ
}
λ>
0
of functions in
L
1
(
R
) such that
(a)
integraldisplay
k
λ
(
x
)
dx
= 1 for every
λ >
0,
(b) sup
λ
bardbl
k
λ
bardbl
1
<
∞
, and
(c) for every
δ >
0,
lim
λ
→∞
integraldisplay

x
≥
δ

k
λ
(
x
)

dx
= 0
.
Note that, by definition, an approximate identity is a family of integrable
functions. If it is the case that
k
λ
≥
0 for each
λ
, then requirement (b) in
Definition 1.51 follows from requirement (a). However, in general the elements
of an approximate identity need not be nonnegative functions.
The “easy” way to create an approximate identity is through dilation of a
single function.
Exercise 1.52.
Let
k
∈
L
1
(
R
) be any function that satisfies
integraldisplay
k
(
x
)
dx
= 1
.
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1.5 Approximate Identities
39
Define
k
λ
by an
L
1
normalized dilation:
k
λ
(
x
) =
λ k
(
λx
)
,
λ >
0
,
and show that the resulting family
{
k
λ
}
λ>
0
forms an approximate identity.
Note that there is an inherent ambiguity in our notation: We may use
{
k
λ
}
λ>
0
to indicate a generic family of functions indexed by
λ
, or, as intro
duced in Notation 1.5, we may use
k
λ
to denote the
L
1
normalized dilation
of a function
k
. The intended meaning is usually clear from context.
In any case, if we define
k
λ
by dilation, then, as
λ
increases,
k
λ
becomes
more and more similar to our intuition of what a “
δ
function” (a function that
is an identity for convolution) would look like (see the illustration in Figure 1.7
and the related discussion in Section 1.3.5). While there is no such identity for
convolution in
L
1
(
R
), the collection of functions
{
k
λ
}
λ>
0
in some sense forms
an approximation to this nonexistent
δ
function, for requirement (c) implies
that
k
λ
becomes more and more concentrated near the origin as
λ
increases,
with
integraltext
k
λ
= 1 for every
λ
.
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 Fall '09
 HEIL
 Fourier Series, approximate identity, lim kλ

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