An equation of convolution type on convex domains in $\mathbf R^2$

Let $D$ be a convex domain in $\mathbf R^2$, and let $C^{(k)}(D)$, $k=(k_1,\,k_2)$, be the space of functions $f(x)$, continuous in $D$ together with their partial derivatives
$$ \frac{\partial^{n_1+n_2}}{\partial x_1^{n_1}\partial x_2^{n_2}}f, $$
$n_1\leqslant k_1$, $n_2\leqslant k_2$. This space is provided with the natural topology of uniform convergence of functions and corresponding derivatives on compact subsets of $D$. Consider in $C^{(k)}(D)$ the homogeneous convolution equation $\mu*f=0$, where $\mu$ is a continuous linear functional on $C^{(k)}(D)$. It is proved that every solution of this equation from the space $C^{(k)}(D)$ can be approximated in the topology of $C^{(k)}(D)$ by a linear combination of exponential polynomials satisfying this equation.
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