10.5. APPROXIMATION WITH SMOOTH FUNCTIONS 253
Proof: If f ≥ 0 then extend it to be 0 off U . In the above argument, let all functionsinvolved, the simple functions and the continuous functions be zero off U . Simply intersectall Vi with U and no harm is done. Now to extend to Lp (U) , simply apply Theorem 10.5.1to the positive and negative parts of real and imaginary parts of f . ■
Example 10.5.4 Let U = B(z,2r)
ψ (x) =
exp[(|x− z|2− r2
)−1]
if |x− z|< r,
0 if |x− z| ≥ r.
Then a little work shows ψ ∈C∞c (U). The following also is easily obtained.
Lemma 10.5.5 Let U be any open set. Then C∞c (U) ̸= /0.
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. ■For a different approach see Problem 13 on Page 213.This leads to a really remarkable result about approximation with smooth functions.
Definition 10.5.6 Let U = {x∈Rn : |x|< 1}. A sequence {ψm} ⊆C∞c (U) is called
a mollifier (This is sometimes called an approximate identity if the differentiability is notincluded.) if
ψm(x)≥ 0, ψm(x) = 0, if |x| ≥ 1m,
and∫
ψm(x) = 1. Sometimes it may be written as {ψε} where ψε satisfies the aboveconditions except ψε (x) = 0 if |x| ≥ ε . In other words, ε takes the place of 1/m. Therecertainly exist mollifiers. Let ψ ∈C∞
c (B(0,1)) , ψ (x)≥ 0,∫
ψ (x)dmn = 1. Then let
ψm (x)≡ cmψ (mx)
where cm is chosen to make∫
cmψ (mx)dmn = 1. Thus ψm is 0 off B(0, 1
m
).
The notation∫
f (x,y)dµ(y) will mean x is fixed and the function y→ f (x,y) is beingintegrated. To make the notation more familiar, dx is written instead of dmn(x).
Lemma 10.5.7 Let g ∈Cc (U) then there exists h ∈C∞c (U) such that∫
|g−h|p dmn < ε.
Proof: Let ψm be a mollifier. Consider
hm (x)≡∫
g(x−y)ψm (y)dmn (y)
Then since the integral of ψm is 1, it follows that
hm (x)−g(x) =∫
(g(x−y)−g(x))ψm (y)dmn (y)