Completeness theorems in the uniform norm for a parabolic equation

Let $\{v_{\alpha}\}$ be a system of polynomial solutions of the parabolic equation $a_{hk}\partial_{x_{h}x_{k}}u - \partial_t u =0$ in a bounded $C^1$-cylinder $\Omega_{T}$ contained in $\mathbb{R}^{n+1}$. Here $a_{hk}\partial_{x_{h}x_{k}}$ is an elliptic operator with real constant coefficients. We prove that $\{v_{\alpha}\}$ is complete in $C^{0}(\Sigma')$, where $\Sigma'$ is the parabolic boundary of $\Omega_{T}$. Similar results are proved for the adjoint equation $a_{hk}\partial_{x_{h}x_{k}} u+ \partial_t u =0$.