Master Dissertation

7.2.2 Uniqueness
We still need to prove that the decomposition of d is unique. To this end
we need the following result
Theorem 7.11. Let u ∈ D ′ (R n ) with support supp u = {0}. Then there
exists a N ∈ N such that
where cα ∈ C.
u =
|a|≤N
cα∂ α δ,
This theorem is a consequence of the following lemmas.
Lemma 7.12.
1
k! ∂k t f(xt) =
|α|=k
x α
α! f (α) (xt) (7.17)
Proof. The first we note that by the chain rule, we may write the left hand
side as
1
k! ∂k t f(xt) = 1
n
k
∂
xi f(y)
k! ∂yi
i=1
y=xt
It is this form we will use when proving (7.17) by induction. That is, we
want to prove
1
n
∂
k xi f(y) =
k! ∂yi
xα α! ∂αf(y) i=1
59
|α|=k