The existence of an analytic solution of an infinite order differential equation and the nature of its domain of analyticity

Yu. F. Korobeinik

Sbornik. Mathematics, Tome 9 (1969) no. 1, pp. 53-71

Résumé

We consider the equation \begin{equation} Ly\equiv\sum_{k=0}^\infty a_k y^{(k)}(z)=f(z) \end{equation} under the assumption that the characteristic function $a(z)=\sum_{k=0}^\infty a_kz^k$ is an entire function which does not grow faster than an exponential function of minimal type (that is, $a(z)\in[1,0]$). If $G$ is an arbitrary domain, we let $E(G)$ denote the set of all functions which are analytic in $G$, and we let $L(E(G))$ be the image of $E(G)$ under the operator $Ly$ acting from $E(G)$ into $E(G)$. We let $W(y)$ denote the complete Weierstrass domain of existence of an arbitrary analytic function $y(z)$. Theorem 1. If $G$ is a finite convex domain, then $L(E(G))= E(G)$. \smallskip Theorem 2. If $G$ is not a simply connected domain, then $L(E(G))$ is a proper subset of $E(G)$. \smallskip Theorem 3. Let the function $y(z)$ be analytic at $z_0\in W(f)$ and satisfy equation $(1)$ in a neighborhood of this point. Then: a) if $W(f)$ is simply connected, then $W(y)$ is simply connected; b) if $W(f)$ is convex, then $W(y)$ is convex. Assertion 3b) for the case where $f(z)$ is an entire function extends a theorem of Polya. We note an important qualitative difference between linear equations of finite and infinite order. Namely, under the assumptions of Theorem 3 for a finite problem we know that $W(y)=W(f)$, but for an infinite problem we can always find a solution $y_1(z)$ for which $W(y_1)$ is a proper subset of $W(f)$. The following theorem is specifically for equations of infinite order, and does not have a finite analog. Theorem 4. {\it If $G$ is a domain which is not convex and $a(z)$ is a transcendental entire function in the class $[1,0],$ then there exists an operator $L_1y=\sum_{k=0}^\infty b_ky^{(k)}(z)$ with characteristic function $a_1=a(e^{i\varphi_2}z),$ $\varphi_2\in[0,2\pi],$ such that $L_1(E(G))$ is a proper subset of $E(G)$}. We note here that if $a(z)$ is a polynomial and $G$ is a finite, simply connected domain, then $L(E(G))=E(G)$. In this work we shall find necessary and sufficient conditions for solvability of equation (1) in $E(G)$ for a given right-hand side $f(z)\in E(G)$. We establish a connection between solvability conditions and certain interpolation problems for exponential functions. We shall examine certain examples. Bibliography: 15 titles.