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

(ϕ ∗ 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 (ϕ 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

(ϕ ∗ 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, (ϕ ∗ 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}.