Boundary properties of analytic solutions of differential equations of infinite order

Yu. F. Korobeinik

Sbornik. Mathematics, Tome 43 (1982) no. 3, pp. 323-345

Résumé

Let $\mathscr L(\lambda)$ be an entire function from the class $[1,0]$ with simple zeros $\{\lambda_n\}$ and let $\mathscr G$ be a bounded convex domain. In this paper particular solutions of the equation \begin{equation} (\mathscr L(D))(z)=f(z),\qquad z\in\mathscr G, \tag{\text{I}} \end{equation} are constructed which are analytic in $\mathscr G$ and possess a definite smoothness on the boundary of $\mathscr G$, for the case in which $f$ is analytic in $\mathscr G$ and sufficiently smooth on the boundary. In particular, it is shown that if $\mathscr L(\lambda)$ is an entire function of completely regular growth with proximate order $\rho(r)$, $\rho(r)\to\rho$, $0\rho1$, with a positive indicator and a regular set of roots, then for an arbitrary function $f$, analytic in $\mathscr G$ and continuous on $\overline{\mathscr G}$, equation (I) has an effectively defined particular solution analytic in $\mathscr G$ and infinitely differentiable at each boundary point of $\mathscr G$. Bibliography: 14 titles.