Mathematics, Physics, and Computer Science Faculty Articles and Research

Title

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

Article

1997

Abstract

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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