22 I. Theory of Distributions
Since |Rm(x)|
C|x|m+1
and β(
x
ε
) = 0 for |x| ε, we have |β(
x
ε
)Rm(x)|
Cεm+1.
Analogously, for any derivative of β(
x
ε
)Rm(x) we get
∂k
∂xk
β
x
ε
Rm(x)
Cεm+1−|k|,
0 |k| m.
Thus
β
x
ε
Rm(x)
m,B1
Cε.
Since f(β(
x)Rm(x))
ε
does not depend on ε, we have that f(β(
x)Rm(x))
ε
= 0.
Therefore
f(ϕ) =
m
|k|=0
dk
∂kϕ(0)
∂xk
=
m
|k|=0
(−1)|k|dk
∂kδ
∂xk
(ϕ).
6.2. Distributions with compact supports.
Denote by E D the space of distributions with compact supports.
Proposition 6.3. Any distribution f can be represented as a limit in D of
distributions fn E .
Proof: Let β(x) be the same as in the proof of Theorem 6.2, and let
fn = β(
x
n
)f. Then supp fn Bn and fn(ϕ) = f(β(
x
n
)ϕ) f(ϕ), ∀ϕ, as
n ∞.
Proposition 6.1 combined with the Hahn-Banach theorem allows one to
prove the following theorem:
Theorem 6.4. Any distribution with a compact support can be represented
in the form
f =
m
|k|=0
∂kfk
∂xk
,
where fk are regular functionals corresponding to the continuous functions
fk(x) with compact supports.
Proof: Let supp f BR. By Proposition 6.1,
(6.12) |f(ϕ)| C|[ϕ]|m1,BR , ∀ϕ C0
∞(BR)
for some m1 0. Denote by ϕ
p
the following norm:
(6.13) ϕ
2
p
=
p
|k|=0
BR
∂kϕ
∂xk
2
dx.
Note that
∂kϕ(x)
∂xk
=
x1
−∞
· · ·
xn
−∞
∂n
∂y1∂y2 · · · ∂yn
∂kϕ(y)
∂yk
dy.
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