26 I. Theory of Distributions
7.2. Convolution of f D and g E .
Definition 7.2. The convolution of f D and g E is defined by the
formula:
(7.9) (f g)(ψ) = f(g1 ψ), ∀ψ D,
where g1 is the distribution obtained from g by the change of variables
x −x (cf. (3.8)):
(7.10) g1(ϕ) = g(ϕ(−x)).
Since g1 has a compact support, we have that g1 ψ = g1(ψ(x ·)) D
(see Proposition 7.1). It follows from (7.3) and Proposition 6.1 that
∂k
∂xk
(g1 ψ) C|[ψ]|m+|k|,BR
for some m and C.
Consider a sequence ψn ψ in D. We assume that supp ψ BR and
supp ψn BR. Then for any k,
max
x∈Rn
∂k
∂xk
(g1 ψn)
∂k
∂xk
(g1 ψ) C|[ψn ψ]|m+|k|,BR 0
as n ∞. By Proposition 7.1, supp(g∗ψn) BR1+R, where supp g1 BR1 .
Therefore g1 ψn g1 ψ in D and f(g1 ψn) f(g1 ψ).
Thus f g is a linear continuous functional on D.
It follows from Proposition 7.2 that Definition 7.2 agrees with Definition
7.1 for g C0
∞.
Proposition 7.3. Let f D and g E . Then for any k = (k1,...,kn),
(7.11)
∂k
∂xk
(f g) =
∂kf
∂xk
g = f
∂kg
∂xk
.
Proof: Indeed
∂k(f
g)
∂xk
(ϕ) =
(−1)|k|(f
g)
∂kϕ
∂xk
=
(−1)|k|f
g1
∂kϕ
∂xk
=
(−1)|k|f
∂k
∂xk
(g1 ϕ) =
∂kf
∂xk
(g1 ϕ) =
∂kf
∂xk
g (ϕ)
and
∂k
∂xk
(f g)(ϕ) =
∂kf
∂xk
(g1 ϕ) =
(−1)|k|f
∂k
∂xk
(g1 ϕ)
=
(−1)|k|f
∂kg1
∂xk
ϕ = f
∂kg
∂xk
(ϕ),
where we used (7.4) and that
∂kg1
∂xk
=
(−1)k
∂kg
∂xk
.
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