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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.

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