On the Approximation of Solutions to the Heat Equation in the Lebesgue Class $$L^2$$ by More Regular Solutions

A criterion for the approximability of all solutions heat equation in a bounded cylindrical domain that belong to Lebesgue class by more regular (e.g., Sobolev) same with larger base is obtained. Namely, complement smaller one must have no (nonempty connected) compact components. As an important corollary, we prove theorem on existence doubly orthogonal basis corresponding pair Hilbert spaces.