On solutions of equations of infinite order in the real domain

A homogeneous partial differential equation of infinite order with constant coefficients of the form
\begin{equation} L[y]\equiv\sum_{|\alpha|\geqslant0}a_\alpha\frac{\partial^{|\alpha|}}{\partial x^\alpha}\,y(x)=0,\qquad\alpha=(\alpha_1,\dots,\alpha_n), \end{equation}
is considered, where $y(x)$ is an infinitely differentiate function that is defined on a convex domain $\Omega\subset R^n$ and satisfies the estimate
$$ \max\biggl|\frac{\partial^{|\alpha|}y(x)}{\partial x^\alpha}\biggr|\leqslant Nh^{|\alpha|}M_{|\alpha|},\qquad N=N(K,y),\quad h=h(K,y), $$
on every compact set $K\Subset\Omega$. It is shown under certain conditions on the sequence $M_{|\alpha|}$ that every solution of equation (1) can be approximated by the exponential solutions of this equation.
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