Holomorphy types and the Fourier-Borel transform between spaces of entire functions of a given type and order defined on Banach spaces

Vinícius V. Fávaro, Ariosvaldo M. Jatobá

Abstract


Let $E$ be a Banach space and $\Theta$ be a $\pi_{1}$-holomorphy type. The main purpose of this paper is to show that the Fourier-Borel transform is an algebraic isomorphism between the dual of the space ${\operatorname{Exp}}_{\Theta,A}^{k}(E)$ of entire functions on $E$ of order $k$ and $\Theta$-type strictly less than $A$ and the space ${\operatorname{Exp}}_{\Theta^{\prime},0,(\lambda (k) A)^{-1}}^{k^{\prime}}(E^{\prime})$ of entire functions on $E^{\prime}$ of order $k^{\prime}$ and $\Theta^{\prime}$-type less than or equal to $(\lambda(k)A)^{-1}$. The same is proved for the dual of the space ${\operatorname{Exp}}_{\Theta,A}^{k}(E)$ of entire functions on $E$ of order $k$ and $\Theta$-type less than or equal to $A$ and the space ${\operatorname{Exp}}_{\Theta^{\prime}, (\lambda(k)A)^{-1}}^{k^{\prime}}( E^{\prime})$ of entire functions on $E^{\prime}$ of order $k^{\prime}$ and $\Theta^{\prime}$-type strictly less than $(\lambda(k)A)^{-1}$. Moreover, the Fourier-Borel transform is proved to be a topological isomorphism in certain cases.

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DOI: http://dx.doi.org/10.7146/math.scand.a-15200

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ISSN 0025-5521 (print) ISSN 1903-1807 (online)

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