Invariant subspaces of analytic functions. III. On the extension of spectral synthesis

Let $f$ be a solution of the equation
\begin{equation*} S*f=0 \end{equation*}
with characteristic function $\varphi$, $D_f$ is the trace which is left by the associated diagram $D$ of the function $\varphi$ under a continuous translational displacement as a geometric figure on the Riemann surface of the function $f$. We show that $D_f$ is a one-sheeted simply connected region; the function $f$ can be uniformly approximated inside $D_f$ by linear combinations of elementary solutions. This result is a corollary of a more general theorem on the extension of spectral synthesis which is proved in this paper.
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