1. Spaces of infinitely differentiable functions 3
With more details in the case of one variable we have
x1 y1 = t1, dt1 = −dy1.
Therefore

−∞
ϕ(x1 y1)ψ(y1)dy1 =
−∞
+∞
ϕ(t1)ψ(x1 t1)dt1
=

−∞
ψ(x1 t1)ϕ(t1)dt1.
b) Assume that ψ
L1(Rn)
and ϕ(x) and
∂kϕ(x)
∂xk
, |k| m, are continu-
ous and bounded. Then ϕ ψ and
∂k
∂xk
ψ), |k| m, are also continuous
and bounded.
Here we use the standard notation : k = (k1,...,kn), |k| = k1 +···+kn,
∂k
∂xk
=
∂k1+k2+···+kn
∂x11
k
∂x22
k
···∂xnnk
.
The proof of property b) is given at the end of the section.
Property b) shows that for the convolution to be smooth it is enough
that one of the two functions or ψ) is smooth.
Proposition 1.2. Let ϕ(x) and ψ(x) be continuous functions with compact
supports, supp ϕ U1, supp ψ U2, where U1 and U2 are open sets in
Rn.
Then
supp(ϕ ψ) U1 + U2,
where U1 + U2 is the open set in
Rn
consisting of all sums x + y, where
x U1 and y U2.
Proof: We have
ψ)(x) =
Rn
ϕ(x y)ψ(y)dy =
U2
ϕ(x y)ψ(y)dy,
since ψ(y) = 0 for y / U2. If x0 / U1 + U2, then x0 y / U1 for all y U2.
Then ϕ(x0 y) = 0 for all y U2 and, therefore, ψ)(x0) = 0. Hence
supp(ϕ ψ) U1 + U2.
1.2. Approximation by C0
∞-functions.
Proposition 1.3. The C0
∞-functions
are dense in the space
Cc(Rn)
of con-
tinuous functions with compact supports: for any continuous ϕ(x) with sup-
port in BR, there exists a sequence of C0
∞-functions
ϕm(x) with support in
BR such that
max
x∈Rn
|ϕ(x) ϕm(x)| 0
as m ∞. Here BR = {x : |x| R}.
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