Holomorphy types and spaces of entire functions of bounded type on banach spaces

In this paper spaces of entire functions of Θ-holomorphy type of bounded type are introduced and results involving these spaces are proved. In particular, we “construct an algorithm” to obtain a duality result via the Borel transform and to prove existence and approximation results for convolution equations. The results we prove generalize previous results of this type due to B. Malgrange: Existence et approximation des équations aux dérivées partielles et des équations des convolutions. Annales de l’Institute Fourier (Grenoble) VI, 1955/56, 271–355; C. Gupta: Convolution Operators and Holomorphic Mappings on a Banach Space, Séminaire d’Analyse Moderne, 2, Université de Sherbrooke, Sherbrooke, 1969; M. Matos: Absolutely Summing Mappings, Nuclear Mappings and Convolution Equations, IMECC-UNICAMP, 2007; and X. Mujica: Aplicações τ (p; q)-somantes e σ(p)-nucleares, Thesis, Universidade Estadual de Campinas, 2006.