Convolution equations and harmonic analysis in spaces of entire functions

Abstract:

If H is the topological space of functions analytic in the simply connected open set $\Omega$ 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 $\Omega$. For f in H and $\phi$ in E, $(f \ast \phi )(z) \equiv \left \langle {f,{\phi _z}} \right \rangle$ where ${\phi _z}$ is the z-translate of $\phi$. If $f{\nequiv }0$ in any component of $\Omega ,f \ast \phi = 0$ if and only if $\phi$ is a finite linear combination of monomial-exponentials ${z^p} \exp (\omega z)$ where $\omega$ is a zero of f in $\Omega$ of order at least $p + 1$. For such f and $\psi$ in E, $f \ast \phi = \psi$ is solved explicitly for $\phi$. If E is assigned its strong dual topology and $\tau (\phi )$ is the closed linear span in E of the translates of $\phi$, then $\tau (\phi )$ 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 $\phi \to f \ast \phi$; there is a one-to-one correspondence between such subspaces and the closed ideals of H with the correspondence that of annihilators.