Invariant subspaces of analytic functions. II.~Spectral synthesis of convex domains

I. F. Krasichkov-Ternovskii

Sbornik. Mathematics, Tome 17 (1972) no. 1, pp. 1-29

Résumé

The criterion for the admissibility of spectral synthesis which was established in the first part of this paper is employed in the solution of a series of problems; in particular, it is employed in the investigation of the homogeneous convolution equation \begin{equation} S*f=0 \tag{\ast} \end{equation} and in the investigation of systems of such equations. Let $H$ be the space of functions holomorphic in a convex region $G$. Let $S$ be a continuous linear functional on $H$. Then the subspace of solutions $f\in H$ of the equation ($\ast$) is invariant and always permits spectral synthesis. However, the system of equations $S_1*f=0,\dots,S_n*f=0$ does not always admit spectral synthesis. In this paper we determine in terms of characteristic functions the precise conditions for the possibility of spectral synthesis for this situation. If $G$ is an unbounded convex region, then spectral synthesis is always possible. Bibliography: 24 titles.