Spectral synthesis on systems of convex domains. Extension of the synthesis

A system of homogeneous convolution equations is considered in convex Domains $G_1,\dots, G_q\subset G$. Earlier (Mat. Sb. (N. S.) 111(153) (1980), 3–41) the author studied the following problem of spectral synthesis: under what conditions can every solution $f=(f_1,\dots,f_q)$ of such a system be approximated by linear combinations of elementary solutions inside $G_1,\dots,G_q$? In the present paper the following problem of the extension of the synthesis is considered: under what conditions does a solution $f=(f_1,\dots,f_q)$ admit approximation not only in $G_1,\dots,G_q$ but also in larger domains $G'_1\supset G_1$, $\dots$, $G'_q\supset G_q$ which are contained in the domains of existence of the components $f_1,\dots,f_q$?
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