The holomorphic Cauchy problem for the convolution operator in analytically uniform spaces, and Fisher expansions. (Q1432593)

An abstract theorem about closed sums of closed subspaces of Fréchet Schwartz spaces and (DFS)-spaces is utilized to derive a criterion to characterize when the Cauchy problem for an abstract surjective convolution operator on an analytically uniform LAU-space in the sense of Hansen, with data on a multiple analytic set, is solvable, respectively well posed. This result is a multi-dimensional analogue of a result due to Napalkov which characterizes the solvability of the holomorphic Cauchy problem for convolution operators on the space \(H(\mathbb{C})\) of entire functions. Several consequences about the Fischer representation of \(H(G)\), \(G\) a convex domain in \(\mathbb{C}^n\), are mentioned.