Convolution equations in spaces of infinite-dimensional entire functions of exponential and related types

Abstract:

We prove results of existence and approximation of the solutions of the convolution equations in spaces of entire functions of exponential type on infinite dimensional spaces. In particular we obtain: let E be a complex, quasi-complete and dual nuclear locally convex space and $\Omega$ a convex balanced open subset of E; let $\mathcal {H} (\Omega )$ be the space of the holomorphic functions on $\Omega$, equipped with the compact open topology and $\mathcal {H}’(\Omega )$ its strong dual; let $\mathcal {F} \mathcal {H}’(\Omega )$ denote the image of $\mathcal {H}’(\Omega )$ through the Fourier-Borel transform $\mathcal {F}$; equip this space $\mathcal {F} \mathcal {H}’(\Omega )$ with the image of the topology of $\mathcal {H}’(\Omega )$ via the map $\mathcal {F}$. Then, “every nonzero convolution operator on $\mathcal {F} \mathcal {H}’(\Omega )$ is surjective” and “every solution of the homogeneous equation is limit of exponential-polynomial solutions". Our results are more generally valid when E is a Schwartz bornological vector space with the approximation property. Previous results in Fréchet-Schwartz and Silva spaces are thus extended to domains that are not Fréchet or D.F.-spaces.