7. The convolution of distributions 27

Proposition 7.4. For any f ∈ E there exists fε ∈ C0

∞(Rn)

such that

fε → f in D as ε → 0.

Proof: Let βε be a delta-like sequence as in Example 4.4, and, in ad-

dition, β(x) ∈ C0

∞(B1).

Let fε(x) = f ∗ βε = f(βε(x − ·)). By (7.6) we

have:

fε(ϕ) = f(βε1 ∗ ϕ),

where βε1(x) = βε(−x). For any k we have

∂k

∂xk

(βε1 ∗ ϕ) = βε1 ∗

∂kϕ

∂xk

→

∂kϕ

∂xk

uniformly (see Proposition 1.3). Since supp(βε1 ∗ ϕ) ⊂ BR+1 for ε 1,

where supp ϕ ⊂ BR (see Proposition 1.2), we have that βε1 ∗ ϕ → ϕ in D.

Therefore fε(ϕ) = f(βε1 ∗ ϕ) → f(ϕ).

Example 7.1. For any f ∈ D we have f ∗ δ = f. Indeed, by (7.9),

(f ∗ δ)(ψ) = f(δ ∗ ψ). We used that δ(−x) = δ(x). By (7.1), δ ∗ ψ =

δ(ψ(x − ·)) = ψ(x). Thus (f ∗ δ)(ψ) = f(ψ) for any ψ ∈ D. Similarly, we

have f ∗

∂kδ

∂xk

=

∂kf

∂xk

.

7.3. Direct product of distributions.

Let f1 ∈

D(Rn1

) and f2 ∈ D

(Rn2

) and let x = (x , x ), where x ∈

Rn1

, x ∈

Rn2

. For any ϕ(x , x ) ∈ C0

∞(Rn1+n2

) the function ϕ1(x ) = f1(ϕ(·,x )) ∈

C0

∞(Rn2

) (cf. the proof of Proposition 7.2). If ϕn(x , x ) → ϕ(x , x ) in

D(Rn1+n2

), then ϕ1n(x ) = f1(ϕn(·,x )) → ϕ1(x ) in

D(Rn2

). Therefore

f2(ϕ1) is well defined, and we call f2(ϕ1) = f2(f1(ϕ)) the direct product

(f1 × f2) ∈ D

(Rn)

of f1 and f2:

(7.12) (f1 × f2)(ϕ) = f2(f1(ϕ(x , x ))), ∀ϕ ∈ C0

∞(Rn1+n2

).

Analogously,

(f2 × f1)(ϕ) = f1(f2(ϕ(x , x ))) ∈

D(Rn1+n2

).

Note that f1 × f2 = f2 × f1. This is obvious in the case where ψ(x , x ) =

∑N

j=1

ψj1(x )ψj2(x ), with ψj1(x ) ∈ C0

∞(Rn1

), ψj2(x ) ∈ C0

∞(Rn2

), since

(f1 × f2)(ψ) =

N

j=1

f1(ψj1)f2(ψf2) = (f2 × f1)(ψ).

For any ϕ ∈ C0

∞(Rn1+n2

), there is a sequence of C0

∞(Rn1+n2

) functions of the

form

∑N

j=1

ψj1(x )ψj2(x ) converging to ϕ(x , x ) in

D(Rn1+n2

). Therefore

we get that (f1 × f2)(ϕ) = (f2 × f1)(ϕ) for all ϕ ∈

D(Rn1+n2

).

Proposition 7.5. Let x = (x , xn) and let f ∈ D

(Rn).

If f(

∂ϕ

∂xn

) = 0 for

any ϕ(x , xn) ∈ C0

∞(Rn),

then f = f1 × 1, where f1 ∈ D

(Rn−1)

and 1 is the

regular functional in D

(R1)

corresponding to 1.