Division Theorems in Spaces of Entire Functions with Growth Conditions and their Applications to PDE of Infinite Order

Abstract

It is shown that every entire function $f\in {\mathrm{Exp}}_0({\mathbf{C}}^n)$, $f(0)\ne 0$ is “slowly decreasing”. As an application of this property, a theorem on analytic continuation of solutions of infinite order differential equations with constant coefficients is proven.