On convolution operators in spaces of entire functions of a given type and order. (English) Zbl 0658.46016

Complex analysis, functional analysis and approximation theory, Proc. Conf., Campinas/Braz. 1984, North-Holland Math. Stud. 125, 129-171 (1986).
[For the entire collection see Zbl 0642.00011.]
Let E be a complex normed space and \({\mathcal K}(E)\) the vector space of all entire functions in E. The author considers several collections of subspaces of \({\mathcal K}(E)\), each of them carrying its own natural locally convex topology. They generalize the spaces associated with A. Martineau.
A convolution equation is a continuous linear mapping from such a space into itself which commutes with all directional derivatives (equivalently, with all translations if all translation operators are defined).
The author studies convolution equations and proves various results on the existence and approximation of the solutions. A basic tool is the Fourier-Borel transformation which representes the duals of the function spaces as other function spaces.
This long paper is well-structured and carefuly written which makes it rather readable in spite of the technical nature of the results.
Reviewer: W.Govaerts


46E10 Topological linear spaces of continuous, differentiable or analytic functions
44A35 Convolution as an integral transform


Zbl 0642.00011