The Fourier–Borel transform between spaces of entire functions of a given type and order

Abstract

In this article we study the Fourier–Borel transform between the dual of ${\mathrm{Exp}}_{\tilde{N},(s;(r,q)),A}^k(E)$, the space of entire functions on $E$ of $(s;(r,q))$-quasi-nuclear order $k$ and $(s;(r,q))$-quasi-nuclear type strictly less than $A$, and the space ${\mathrm{Exp}}_{(s^{\prime },m(r^{\prime };q^{\prime })),0,(\theta (k)A{)}^{-1}}^{k^{\prime }}(E^{\prime })$ of entire functions on $E^{\prime }$ of $(s^{\prime },m(r^{\prime };q^{\prime }))$-summing order $k$ and $(s^{\prime },m(r^{\prime };q^{\prime }))$-summing type less than or equal to $(\theta (k)A{)}^{-1}$. This mapping identifies algebraically and topologically these two spaces. On the dual space it is considered the strong topology. This generalizes results of Matos [4] and Martineau [3].