On the “Grouping” Phenomenon for Holomorphic Solutions of Infinite Order Differential Equations

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Purpose of this paper is to clarify some interpolation questions related to the space $\mathcal{H}'(\Omega)$ of analytic functionals carried by $\mathrm{a}\cdot \mathrm{C}\mathrm{o}\mathrm{m}_{\mathrm{P}^{\mathrm{a}\mathrm{c}\mathrm{t}}}$ set $K$ contained in the convex open set $\Omega\subseteq C$ . In particular, I will address the issue of”grouping of terms” which arises when one looks for exponential series representations of solutions, in the space $\mathcal{H}(\Omega)$ of holomorphic function..s on $\Omega,$ . of $\mathrm{s}\mathrm{y}_{\mathrm{S}\mathrm{t}\mathrm{e}}\mathrm{m}.\mathrm{S}$ of infinite order differential equations. Virtually all of the material contained in this paper originates from [$2|,$ [$5|$ and [6] (but see also the recent monograph [1]); on the other hand, we are presenting this material in a unified fashion as a background to a joint work with T. Kawai [7] on the application of these ideas to some very classical overconvergence results ([3], [4] $)$ . Of some interest, hopefully, is the final construction of a large class of examples of infinite order differential $\mathrm{o}\mathrm{p}\mathrm{e}\mathrm{r}\dot{\mathrm{a}}$tors for which ”grouping” is necessary.


This article was originally published in Res. Inst. Math. Sci. Kokyuroku, volume 1001, in 1997.

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