Completeness theorems on the boundary for a parabolic equation

Let $\{v_{\alpha }\}$ be a system of polynomial solutions of the parabolic equation $a_{hk}{\mathrm{\partial }}_{x_h{x}_k}u-{\mathrm{\partial }}_{t}u=0$ in a bounded $C^1$-cylinder ${\mathrm{\Omega }}_T$ contained in ${\mathbb{R}}^{n+1}$. Here $a_{hk}{\mathrm{\partial }}_{x_h{x}_k}$ is an elliptic operator with real constant coefficients. We prove that $\{v_{\alpha }\}$ is complete in $L^p({\mathrm{\Sigma }}^{\mathrm{\prime }})$, where ${\mathrm{\Sigma }}^{\mathrm{\prime }}$ is the parabolic boundary of ${\mathrm{\Omega }}_T$. Similar results are proved for the adjoint equation $a_{hk}{\mathrm{\partial }}_{x_h{x}_k}u+{\mathrm{\partial }}_{t}u=0$.