If H is the topological space of functions analytic in the simply connected open set
Ω
\Omega
of the plane with the topology of compact convergence, its dual may be identified with the space E of functions of exponential type whose Borel transforms have their singularities in
Ω
\Omega
. For f in H and
ϕ
\phi
in E,
(
f
∗
ϕ
)
(
z
)
≡
⟨
f
,
ϕ
z
⟩
(f \ast \phi )(z) \equiv \left \langle {f,{\phi _z}} \right \rangle
where
ϕ
z
{\phi _z}
is the z-translate of
ϕ
\phi
. If
f
≢
0
f{\nequiv }0
in any component of
Ω
,
f
∗
ϕ
=
0
\Omega ,f \ast \phi = 0
if and only if
ϕ
\phi
is a finite linear combination of monomial-exponentials
z
p
exp
(
ω
z
)
{z^p} \exp (\omega z)
where
ω
\omega
is a zero of f in
Ω
\Omega
of order at least
p
+
1
p + 1
. For such f and
ψ
\psi
in E,
f
∗
ϕ
=
ψ
f \ast \phi = \psi
is solved explicitly for
ϕ
\phi
. If E is assigned its strong dual topology and
τ
(
ϕ
)
\tau (\phi )
is the closed linear span in E of the translates of
ϕ
\phi
, then
τ
(
ϕ
)
\tau (\phi )
is a finite direct sum of closed subspaces spanned by monomial-exponentials. Each closed translation invariant subspace of E is the kernel of a convolution mapping
ϕ
→
f
∗
ϕ
\phi \to f \ast \phi
; there is a one-to-one correspondence between such subspaces and the closed ideals of H with the correspondence that of annihilators.