but
even more incredible things can be said. In fact, you can approximate an arbitrary function in
L^{1}
(ℝp )
with one which is infinitely differentiable having compact support. From now on, the
notation
|x|
will refer to the Euclidean norm,
(∑n )1∕2
|x | = |xk|2 .
k=1
This is very important in partial differential equations. I am just giving a short introduction
to this concept here. Consider the following example.
Example 10.1.1Let U = B
(z,2r)
( [ ]
{ exp (|x− z|2 − r2) −1 if |x − z| < r,
ψ (x) =
( 0 if |x − z| ≥ r.
Then a little work shows ψ ∈ C_{c}^{∞}(U). Also note that ifz = 0,then ψ
(− x)
= ψ
(x)
. Thismeans it has all derivatives and vanishes off a compact set. The following also is easilyobtained.
You show this by verifying the partial derivatives all exist and are continuous. The only
place this is hard is when
|x− z|
= r. It is left as an exercise. You might consider a simpler
example,
{
e− 1∕x2 if x ⁄= 0
f (x) = 0 if x = 0
and reduce the above to a consideration of something like this simpler case.
Lemma 10.1.2Let U be any open set. Then C_{c}^{∞}(U)≠∅.
Proof: Pick z ∈ U and let r be small enough that B
(z,2r)
⊆ U. Then let
ψ ∈ C_{c}^{∞}
(B (z,2r))
⊆ C_{c}^{∞}
(U )
be the function of the above example. ■
Definition 10.1.3Let U = {x ∈ ℝ^{p} : |x| < 1}. A sequence {ψ_{m}}⊆ C_{c}^{∞}(U) iscalled amollifier if
1-
ψm (x) ≥ 0,ψm (x) = 0, if |x| ≥ m ,
and∫ψ_{m}(x) = 1. Sometimes it may be written as
{ψε}
where ψ_{ε}satisfies the aboveconditions except ψ_{ε}
(x)
= 0 if
|x|
≥ ε. In other words, ε takes the place of 1∕m and ineverything that follows ε → 0 instead of m →∞.
∫f(x,y)dm_{p}(y) will mean x is fixed and the function y → f(x,y) is being integrated. To
make the notation more familiar, dx is written instead of dm_{p}(x).
Example 10.1.4Let
∞
ψ ∈ Cc (B (0,1))(B(0,1) = {x : |x| < 1})
with ψ(x) ≥ 0 and∫ψdm = 1. Let ψ_{m}(x) = c_{m}ψ(mx) where c_{m}is chosen in such a way that∫ψ_{m}dm = 1.
Definition 10.1.5A function, f, is said to be in L_{loc}^{1}(ℝ^{p})if fis Lebesguemeasurable and if |f|X_{K}∈ L^{1}(ℝ^{p})for every compact set, K.If f ∈ L_{loc}^{1}(ℝ^{p}), andg ∈ C_{c}(ℝ^{p}),
∫
f ∗g(x) ≡ f(y)g(x − y)dy.
This is called the convolution of f and g.
The following is an important property of the convolution.
Proposition 10.1.6Let f and g be as in the above definition. Then
∫ ∫
f(y)g(x − y)dy = f (x − y)g(y )dy
Proof: This follows right away from the change of variables formula. In the left, let
x − y ≡ u. Then the left side equals
∫
f (x − u)g(u)du
because the absolute value of the determinant of the derivative is 1. Now replace u with y and
this proves the proposition. ■
The following lemma will be useful in what follows. It says among other things that one of
these very unregular functions in L_{loc}^{1}
(ℝp)
is smoothed out by convolving with a
mollifier.
Lemma 10.1.7Let f ∈ L_{loc}^{1}(ℝ^{p}), and g ∈ C_{c}^{∞}(ℝ^{p}). Then f ∗ g is an infinitelydifferentiable function. Also, if
{ψm }
is a mollifier and U is an open set andf ∈ C^{0}
(U )
∩ L_{loc}^{1}
p
(ℝ )
, then at every x ∈ U,
lmi→m∞ f ∗ψm (x) = f (x).
If f ∈ C^{1}
(U )
∩ L_{loc}^{1}
(ℝp)
and x ∈ U,
(f ∗ψm ) (x) = fxi ∗ψm (x).
xi
Also, if f ∈ C_{c}
(U)
then f ∗ ψ_{m}→ f uniformly.
Proof:Consider the difference quotient for calculating a partial derivative of
f ∗ g.
∫
f-∗g-(x-+-tej)-− f-∗g-(x-)= f(y)g(x-+-tej −-y-)−-g(x−-y)dy.
t t
Using the fact that g ∈ C_{c}^{∞}
(ℝp)
, the quotient,
g(x-+-tej −-y)−-g(x-− y-),
t
is uniformly bounded. To see this easily, use Theorem 5.5.2 on Page 279 to get the existence
of a constant, M depending on
max {||Dg (x)|| : x ∈ ℝp}
such that
|g(x + tej − y)− g(x− y )| ≤ M |t|
for any choice of x and y. Therefore, there exists a dominating function for the integrand of
the above integral which is of the form C
|f (y)|
X_{K} where K is a compact set
depending on the support of g. It follows from the dominated convergence theorem
the limit of the difference quotient above passes inside the integral as t → 0 and
so
∂ ∫ ∂
---(f ∗g)(x) = f(y )---g(x− y )dy.
∂xj ∂xj
Now letting
-∂-
∂xj
g play the role of g in the above argument, a repeat of the above reasoning
shows partial derivatives of all orders exist. A similar use of the dominated convergence
theorem shows all these partial derivatives are also continuous.
It remains to verify the claim about the mollifier. Let x ∈ U and let m be large enough
that B
( )
x,m1
⊆ U. Then
∫
|f ∗ g(x)− f (x)| ≤ |f (x − y )− f (x)|ψm (y )dy
B(0,1m)
By continuity of f at x, for all m sufficiently large, the above is dominated by
∫
ε ψm (y)dy = ε
B (0,1m)
and this proves the claim.
Now consider the formula in the case where f ∈ C^{1}
f ∗ψm (x + hei)− f ∗ψm (x)
------------h------------ =
1 (∫ ∫ )
-- 1 f (x+hei− y)ψm (y)dy− 1 f (x − y)ψm (y)dy
h B (0,m) B(0,m)
∫
= (f (x+hei-− y)−-f-(x−-y))ψ (y)dy
B(0,1m) h m
Now letting m be small enough and using the continuity of the partial derivatives, it
follows the difference quotients are uniformly bounded for all h sufficiently small
and so one can apply the dominated convergence theorem and pass to the limit
obtaining
∫
ℝp fxi (x − y)ψm (y)dy ≡ fxi ∗ ψm (x )
Finally, consider the last claim.
|∫ | ∫
|| f (x− y)ψ (y)dy − f (x)||≤ |f (x − y)− f (x)|ψ (y)dy
| U m | U m
By uniform continuity of f for m large enough,
|f (x− y) − f (x)|
< ε for all x
whenever
|y |
< 1∕m and so for all m large enough, the above integral is dominated
by
∫
∥f ∗ψm − f ∥ ≤ εψm (y)dy = ε. ■
B(0,1∕m)
Theorem 10.1.8Let K be a compact subset of an open set, U. Then there exists afunction, h ∈ C_{c}^{∞}(U), such that h(x) = 1 for all x ∈ K and h(x) ∈ [0,1] for all x. Also thereexists an open set W such that
---
K ⊆ W ⊆ W ⊆ U
such that Wis compact.
Proof: Let r > 0 be small enough that K + B(0,3r) ⊆ U. The symbol, K + B(0,3r)
means
{k + x : k ∈ K and x ∈ B(0,3r)}.
Thus this is simply a way to write
∪{B (k,3r) : k ∈ K} .
Think of it as fattening up the set, K. Let K_{r} = K + B(0,r). A picture of what is
happening follows. Consider X_{Kr}∗ ψ_{m} where ψ_{m} is a mollifier. Let m be so large
PICT
that
1m-
< r. Then from the definition of what is meant by a convolution, and using that ψ_{m}
has support in B
( )
0, 1m
, X_{Kr}∗ψ_{m} = 1 on K and its support is in K + B
(0,3r)
, a bounded
set. Now using Lemma 10.1.7, X_{Kr}∗ ψ_{m} is also infinitely differentiable. Therefore, let
h = X_{Kr}∗ ψ_{m}.
As to the existence of the open set W, let it equal the closed and bounded set h^{−1}
([ ])
12,1
.
This proves the theorem. ■
The following is the remarkable theorem mentioned above. First, here is some
notation.
Definition 10.1.9Let g be a function defined on a vector space. Theng_{y}
(x )
≡ g
(x − y)
.
Theorem 10.1.10C_{c}^{∞}(ℝ^{p}) is dense in L^{1}(ℝ^{p}). By this is meant that for everyf ∈ L^{1}
p
(ℝ )
and ε > 0, there exists g ∈ C_{c}^{∞}
p
(ℝ )
such that
∫
|f − g|dmp < ε
ℝp
Proof: Let f ∈ L^{1}(ℝ^{p}) and let ε > 0 be given. Choose g ∈ C_{c}(ℝ^{p}) such that
whenever m is large enough. This follows because since g has compact support, it is
uniformly continuous on ℝ^{p} and so if η > 0 is given, then whenever
|y|
is sufficiently
small,
|g (x) − g(x− y)| < η
for all x. Thus, since g has compact support, if y is small enough, it follows
∫
|g − gy|dmp (x ) < ε∕2.
There is no measurability problem in the use of Fubini’s theorem because the function
Another important application of Theorem 10.1.8 has to do with a partition of
unity.
Definition 10.1.11A collection of sets ℋ is called locally finite if for everyx,there exists r > 0 such that B
(x,r)
has nonempty intersection with only finitely manysets of ℋ.Of course every finite collection of sets is locally finite. This is the case ofmost interest in this book but the more general notion is interesting.
The thing about locally finite collection of sets is that the closure of their union equals the
union of their closures. This is clearly true of a finite collection.
Lemma 10.1.12Let ℋ be a locally finite collection of sets of a normed vector space V .Then
--- { -- }
∪ ℋ = ∪ H : H ∈ ℋ .
Proof: It is obvious ⊇ holds in the above claim. It remains to go the other way. Suppose
then that p is a limit point of ∪ℋ and p
∕∈
∪ℋ. There exists r > 0 such that B
(p,r)
has
nonempty intersection with only finitely many sets of ℋ say these are H_{1},
⋅⋅⋅
,H_{m}. Then
I claim pmust be a limit point of one of these. If this is not so, there would exist r^{′} such
that 0 < r^{′}< r with B
(p,r′)
having empty intersection with each of these H_{i}. But
then p would fail to be a limit point of ∪ℋ. Therefore, p is contained in the right
side. It is clear ∪ℋ is contained in the right side and so This proves the lemma.
■
A good example to consider is the rational numbers each being a set in ℝ. This is
not a locally finite collection of sets and you note that ℚ = ℝ≠∪
{x-: x ∈ ℚ}
. By
contrast, ℤ is a locally finite collection of sets, the sets consisting of individual integers.
The closure of ℤ is equal to ℤ because ℤ has no limit points so it contains them
all.
Notation 10.1.13I will write ϕ ≺ V to symbolize ϕ ∈ C_{c}
(V)
, ϕ has values in
[0,1]
,and ϕ has compact support in V . I will write K ≺ ϕ ≺ V for K compact and V opento symbolize ϕ is 1 on K and ϕ has values in
[0,1]
with compact support contained inV .
A version of the following lemma is valid for locally finite coverings, but we are only using
it when the covering is finite.
Lemma 10.1.14Let K be a closed set in ℝ^{p}and let
{Vi}
_{i=1}^{n}be a finite list of boundedopen sets whose union contains K. Then there exist functions, ψ_{i}∈ C_{c}^{∞}
(Vi)
such that for allx ∈ K,
∑n
1 = ψi(x)
i=1
and the function f
(x)
given by
∑n
f (x) = ψi(x)
i=1
is in C^{∞}
(ℝp)
.
Proof: Let K_{1} = K ∖∪_{i=2}^{n}V_{i}. Thus K_{1} is compact because K_{1}⊆ V_{1}. Let W_{1} be an
open set having compact closure which satisfies
---
K1 ⊆ W1 ⊆ W 1 ⊆ V1
Thus W_{1},V_{2},
⋅⋅⋅
,V_{n} covers K and W_{1}⊆ V_{1}. Suppose W_{1},
⋅⋅⋅
,W_{r} have been
defined such that W_{i}⊆ V_{i} for each i, and W_{1},
⋅⋅⋅
,W_{r},V_{r+1},
⋅⋅⋅
,V_{n} covers K. Then
let
( n ) ( r )
Kr+1 ≡ K ∖( ∪i=r+2Vi ∪ ∪ j=1Wj ).
It follows K_{r+1} is compact because K_{r+1}⊆ V_{r+1}. Let W_{r+1} satisfy
--- ---
Kr+1 ⊆ Wr+1 ⊆ W r+1 ⊆ Vr+1,W r+1 is compact
Continuing this way defines a sequence of open sets
{Wi}
_{i=1}^{n} having compact closures with
the property
---
Wi ⊆ Vi,K ⊆ ∪ni=1Wi.
Note
{Wi }
_{i=1}^{n} is locally finite because the original list,
{Vi}
_{i=1}^{n} was locally finite. Now let
U_{i} be open sets which satisfy
--- -- --
W i ⊆ Ui ⊆ U i ⊆ Vi,U i is compact.
Similarly,
{Ui}
_{i=1}^{n} is locally finite.
PICT
Now ∪_{i=1}^{n}W_{i} = ∪_{i=1}^{n}W_{i} and so it is possible from Theorem 10.1.8 to define ϕ_{i} and γ,
infinitely differentiable functions having compact support such that
∪_{i=1}^{n}U_{i} because ϕ_{i} equals one on U_{i}.
Consequently γ
(y)
= 0 for all y near x thanks to the fact that ∪_{i=1}^{n}U_{i} is closed and so
ψ_{i}(y) = 0 for all y near x. Hence ψ_{i} is infinitely differentiable at such x. If ∑_{j=1}^{n}ϕ_{j}(x