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

 

Division by holomorphic functions and convolution equations in infinite dimension


Authors: J.-F. Colombeau, R. Gay and B. Perrot
Journal: Trans. Amer. Math. Soc. 264 (1981), 381-391
MSC: Primary 46F25; Secondary 32A15, 46G20
MathSciNet review: 603769
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Abstract: Let $ E$ be a complex complete dual nuclear locally convex space (i.e. its strong dual is nuclear), $ \Omega $ a connected open set in $ E$ and $ \mathcal{E}(\Omega )$ the space of the $ {C^\infty }$ functions on $ \Omega $ (in the real sense). Then we show that any element of $ \mathcal{E}'(\Omega )$ may be divided by any nonzero holomorphic function on $ \Omega $ with the quotient as an element of $ \mathcal{E}'(\Omega )$. This result has for standard consequence a new proof of the surjectivity of any nonzero convolution operator on the space $ \operatorname{Exp} (E')$ of entire functions of exponential type on the dual $ E'$ of $ E$. As an application of the above division result and of a result of $ {C^\infty }$ solvability of the $ \overline \partial $ equation in strong duals of nuclear Fréchet spaces we study the solutions of the homogeneous convolution equations in $ \operatorname{Exp} (E')$ in terms of the zero set of their characteristic functions.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1981-0603769-9
PII: S 0002-9947(1981)0603769-9
Keywords: Infinite dimensional holomorphy, infinite dimensional distribution, convolution equation, $ \overline \partial $ equation, Fourier-Borel transform
Article copyright: © Copyright 1981 American Mathematical Society