2 I. Theory of Distributions
Laplace equations the traditional approach is simpler, and we offer this
approach as problems (with some hints).
1. Spaces of infinitely differentiable functions
Denote by C0
the space of infinitely differentiable complex-valued func-
with compact supports, i.e., ϕ(x) ∈ C0
if ϕ(x) has contin-
uous partial derivatives of every order and ϕ(x) = 0 when |x| R for some
R depending of ϕ(x). For example,
(1.1) χ(x) =
1−|x|2 if |x| 1,
0 if |x| ≥ 1, |x| = x1
+ · · · + xn,2
is a C0
Definition 1.1. Let f(x) be a continuous function in
The support of
f(x) is the closure of the set where f(x) = 0.
We denote the support of f(x) by supp f.
a) The support of χ(x) is the closed ball of radius 1.
b) The support of f(x1) = x1
− 1 is
Definition 1.2. Let ϕ(x) be a measurable bounded and ψ(x) a Lebesgue
integrable functions in
|ψ(x)|dx +∞. Then the convo-
lution of ϕ(x) and ψ(x) (denoted by (ϕ ∗ ψ)(x) ) is the following integral:
(1.2) (φ ∗ ψ)(x) =
φ(x − y)ψ(y)dy.
Proposition 1.1. If ϕ(x) ∈ L1 and ψ(x) ∈ L1, then the integral (1.2) exists
for almost all x ∈
ϕ ∗ ψ ∈
(1.3) ϕ ∗ ψ
The proof of Proposition 1.1 is given at the end of this section.
1.1. Properties of the convolution.
a) ϕ ∗ ψ = ψ ∗ ϕ.
Proof: We have
(1.4) (ϕ ∗ ψ)(x) =
ϕ(x − y)ψ(y)dy.
Changing the variables x − y = t, we get
ϕ(x − y)ψ(y)dy =
ϕ(t)ψ(x − t)dt = ψ ∗ ϕ.