24 I. Theory of Distributions
Remark 6.1. Let g D (BR ), where BR BR (cf. the definition of D (Ω)
in §2). Then, as in (6.17), the restriction of g to BR can be represented in
the form
g(ϕ) =
m
|k|=0
∂kgk
∂xk
(ϕ), ∀ϕ C0
∞(BR),
where the gk are regular functionals in D (BR) corresponding to the func-
tions gk(x) continuous in BR.
7. The convolution of distributions
7.1. Convolution of f D and ϕ C0
∞.
If ϕ C0
∞(Rn),
then, obviously, ϕ(x y) C0
∞(Rn)
as a function of y for
any fixed x
Rn.
Definition 7.1. We define the convolution of a distribution f D and a
function ϕ D by the formula:
(7.1) (f ϕ)(x) = f(ϕ(x ·)),
where f acts on ϕ(x y) as a function of y.
Since ϕ(xm −y) ϕ(x0 −y) in D as xm x0 and since f is a continuous
functional, we have that (f ϕ)(x) is a continuous function of x. Also we
have that
(7.2)
ϕ(x y , xn + hn yn) ϕ(x y , xn yn)
hn
=
∂ϕ(x y , xn + θhn yn)
∂xn

∂ϕ(x y , xn yn)
∂xn
in D,
as hn 0 and (x , xn) is fixed. Here x = (x1,...,xn−1),y = (y1,...,yn−1).
Therefore, the limit
∂(f ϕ)
∂xn
= lim
Δxn→0
(f ϕ)(x , xn + Δxn) (f ϕ)(x , xn)
Δxn
= lim
Δxn→0
f
∂ϕ(x ·,xn + θΔxn ·)
∂xn
exists and
∂(f ϕ)
∂xn
= f
∂ϕ(x ·)
∂xn
.
Analogously, one can show that each partial derivative
∂k(f∗ϕ)
∂xk
exists and is
continuous, and
(7.3)
∂k(f
ϕ)
∂xk
= f
∂kϕ(x
·)
∂xk
= f
∂kϕ
∂xk
.
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