6 I. Theory of Distributions
Let h(x) = ψ)(x). Then
∂h(x , xn)
∂xn
= lim
Δxn→0
h(x , xn + Δxn) h(x , xn)
Δxn
= lim
Δxn→0
Rn
ϕ(x y , xn + Δxn yn) ϕ(x y , xn yn)
Δxn
ψ(y , yn)dy dyn,
where x = (x1,...,xn−1), y = (y1,...,yn−1).
By the mean value theorem,
ϕ(x y , xn + Δxn yn) ϕ(x y , xn yn)
Δxn
=
∂ϕ(x y , xn + θΔxn yn)
∂xn
,
where 0 θ 1. Since
∂ϕ
∂xn
is bounded and continuous, we have that
|
∂ϕ(x −y ,xn+θΔxn−yn)
∂xn
| M and
lim
Δxn→0
∂ϕ(x y , xn + θΔxn yn)
∂xn
=
∂ϕ(x y , xn yn)
∂xn
.
Therefore, by the Lebesgue convergence theorem,
∂h(x , xn)
∂xn
=
∂ϕ
∂xn
ψ
and, as was already proven,
∂h
∂xn
is continuous since
∂ϕ
∂xn
is continuous.
Analogously one treats partial derivatives of any order.
2. Definition of a distribution
Definition 2.1. A linear continuous functional f on D is called a distribu-
tion. Thus f(ϕ) : D C is a distribution if the following two properties
hold:
1. Linearity: f(α1ϕ1 + α2ϕ2) = α1f(ϕ1) + α2f(ϕ2) for any αk
C, ϕk D, k = 1, 2.
2. Continuity: If ϕn ϕ in D, then f(ϕn) f(ϕ).
2.1. Examples of distributions.
a) Let f(x) be a locally absolutely integrable function, i.e.,
BR
|f(x)|dx +∞
for any R 0. (We assume that f(x) is Lebesgue integrable.) Define
(2.1) f(ϕ) =
Rn
f(x)ϕ(x)dx, ∀ϕ D.
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