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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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


Authors: J.-F. Colombeau and B. Perrot
Journal: Trans. Amer. Math. Soc. 258 (1980), 191-198
MSC: Primary 46G20; Secondary 32A15, 35R15
MathSciNet review: 554328
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Abstract: 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 $ \Omega $ a convex balanced open subset of E; let $ \mathcal{H} (\Omega )$ be the space of the holomorphic functions on $ \Omega $, equipped with the compact open topology and $ \mathcal{H}'(\Omega )$ its strong dual; let $ \mathcal{F} \mathcal{H}'(\Omega )$ denote the image of $ \mathcal{H}'(\Omega )$ through the Fourier-Borel transform $ \mathcal{F} $; equip this space $ \mathcal{F} \mathcal{H}'(\Omega )$ with the image of the topology of $ \mathcal{H}'(\Omega )$ via the map $ \mathcal{F} $. Then, ``every nonzero convolution operator on $ \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.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1980-0554328-7
PII: S 0002-9947(1980)0554328-7
Keywords: Infinite-dimensional holomorphy, convolution equations, Fourier-Borel transform, bornology
Article copyright: © Copyright 1980 American Mathematical Society