22 I. Theory of Distributions
Since |Rm(x)| ≤
) = 0 for |x| ε, we have |β(
Analogously, for any derivative of β(
)Rm(x) we get
0 |k| ≤ m.
does not depend on ε, we have that f(β(
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(
)f. Then supp fn ⊂ Bn and fn(ϕ) = f(β(
)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
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
for some m1 ≥ 0. Denote by ϕ
the following norm:
· · ·
∂y1∂y2 · · · ∂yn