The Stieltjes convolution and a functional calculus for non-negative ...

This shows that the distribution in the following Definition 8 is well-defined
and in D S (R + ) ′ .
Definition 8 For two distributions S,T ∈ D S (R + ) ′ we define the convolution
S ∗ T ∈ D S (R + ) ′ by
(S ∗ T)(ϕ) := ( (S ◦ τ) ∗ (T ◦ τ) ) (τ −1 ϕ) ∀ϕ ∈ D(R).
Lemma 5 (i) The extended Stieltjes convolution in Definition 8 is commutative,
associative.
(ii) ∀S,T ∈ D S (R + ) ′ : S(S ∗ T) = S(S) · S(T)
Proof: Commutativity follows directly from the definition and the commutativity
of the convolution on D S ∗(R + ) ′ . For the last property we use representation
(20) for the convolution on D S ∗(R + ) ′ and derive easily the analogous
relation for all functions ϕ ∈ G(R + ) and distributions S,T ∈ D S (R + ) ′ with
measures µ i k i
from Theorem 2. Following the lines of the proof in Lemma 4 we
see that also in this general case we have S(S ∗T) = S(S)·S(T) and therefore
associativity. ✷
The representation formula (20) for the convolution on D S (R + ) ′ also shows
that the convolution defined in Definition 8 coincides with the one from Definition
7 in the case that both distributions S and T are in D S ∗(R + ) ′ .
We now give a brief counterexample similar to the one mentioned in the introduction
that shows that there are measures so that the product of their
Stieltjes transforms is not the Stieltjes transform of any other measure.
Example 1 δ 0 ∗ δ 0 = Dδ 0 .
Proof: One can see that δ 0 is in D S (R + ) ′ but not in D S ∗(R + ) ′ . Its Stieltjes
transform is
∫
1
S(δ 0 )(a) = dδ 0 (λ)
R + λ + a = 1 a
for ∀a > 0. Therefore we find that
∫
S(δ 0 ∗ δ 0 )(a) = S(δ 0 )(a) · S(δ 0 )(a) = 1 = 1! dδ
a 2 0 (λ) ( ) 1+1
1
λ+a = S(Dδ0 )(a)
R +
for ∀a > 0, which shows that δ 0 ∗ δ 0 = Dδ 0 .
✷
Finally, we want to state that the convolution algebra does not have a neutral
element I since this would need to fulfill S ∗ I = S for ∀S ∈ D S ∗(R n +) ′ and
therefore S(S) = S(S) · S(I) ⇒ S(I) ≡ 1. This is not possible because every
Stieltjes transform tends to 0 with ⃗a → ∞.
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