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 Ω \Omega a convex balanced open subset of E; let H ( Ω ) \mathcal {H} (\Omega ) be the space of the holomorphic functions on Ω \Omega , equipped with the compact open topology and H ′ ( Ω ) \mathcal {H}’(\Omega ) its strong dual; let F H ′ ( Ω ) \mathcal {F} \mathcal {H}’(\Omega ) denote the image of H ′ ( Ω ) \mathcal {H}’(\Omega ) through the Fourier-Borel transform F \mathcal {F} ; equip this space F H ′ ( Ω ) \mathcal {F} \mathcal {H}’(\Omega ) with the image of the topology of H ′ ( Ω ) \mathcal {H}’(\Omega ) via the map F \mathcal {F} . Then, “every nonzero convolution operator on F H ′ ( Ω ) \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.