On approximation of solutions to the heat equation from Lebesgue class $L^2$ by more regular solutions

Let $s \in {\mathbb N}$, $T_1,T_2 \in {\mathbb R}$, $T_1<T_2$, and $\Omega, \omega$ be bounded domains in ${\mathbb R}^n$, $n \geq 1$, such that $\omega \subset \Omega$ and the complement $\Omega \setminus \omega$ has no (non-empty) compact components in $\Omega$. We prove that this is the necessary and sufficient condition for the space $H^{2s,s} _{\mathcal H} (\Omega \times (T_1,T_2))$ of solutions to the heat operator ${\mathcal H}$ in a cylinder domain $\Omega \times (T_1,T_2)$ from the anisotropic Sobolev space $H^{2s,s} (\Omega \times (T_1,T_2))$ to be dense in the space $L^{2} _{\mathcal H}(\omega \times (T_1,T_2))$, consisting of solutions in the domain $\omega \times (T_1,T_2)$ from the Lebesgue class $L^{2} (\omega \times (T_1,T_2))$. As an important corollary we obtain the theorem on the existence of a basis with the double orthogonality property for the pair of the Hilbert spaces $H^{2s,s} _{\mathcal H} (\Omega \times (T_1,T_2))$ and $L^{2} _{\mathcal H}(\omega \times (T_1,T_2))$ .