Spectral synthesis in the kernel of a convolution operator in weighted spaces

Weighted spaces of analytic function on a bounded convex domain $D\subset \mathbb C^p$ are treated. Let $U =\{u_n\}_{n=1}^\infty$ be a monotone decreasing sequence of convex functions on $D$ such that $u_n(z)\to\infty$ as $\operatorname{dist}(z,\partial D)\to 0$. The symbol $H(D,U)$ stands for the space of all $f\in H(D)$ satisfying $|f(z)|\exp(-u_n(z))\to 0$ as $\operatorname{dist}(z,\partial D)\to 0$, for all $n\in \mathbb N$. This space is endowed with a locally convex topology with the aid of the seminorms $p_n(f)=\sup\limits_{z\in D}|f(z)|\exp(-u_n(z))$, $n=1,2,\dots$ . Clearly, every functional $S\in H^*(D)$ is a continuous linear functional on $H(D,U)$, and the corresponding convolution operator $M_S\colon f\to S_w(f(z+w))$ acts on $H(D,U)$. All elementary solutions of the equation $M_S[f]=0\ (*)$, i.e., all solutions of the form $z^\alpha e^{\langle a,z\rangle}$, $\alpha\in\mathbb Z_+^p$, $a\in\mathbb C^p$, belong to $H(D,U)$. It is shown that the system $E(S)$ of elementary solutions is dense in the space of solutions of equation $(*)$ that belong to $H(D,U)$.