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

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.
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