Duhamel product in spaces of entire functions of exponential type

Abstract

Let Ω be a polydomain in CN containing the point 0; H(Ω) be the space of all holomorphic functions on Ω with the compact open topology. In the dual H(Ω)' of H(Ω) a multiplication ⊛ is introduced and investigated. It is defined by the convolution rule with the help of shifts which are associated with the system of partial backward shift operators. The realizations of the algebra (H(Ω)',⊛) with the help of the Cauchy and the Laplace transformations and its representation in H(Ω) are obtained. By means of the Laplace transformation the introduced multiplication ⊛ is realized as the many-dimensional Duhamel product in the appropriate space of entire functions of exponential type.