Application of the Stone-Weierstrass theorem to the creation of complete sequences in spaces of integrable functions (Q1323183)

Let \(D \subseteq \mathbb{R}^ m\) be any domain and let \(P\) be any positive definite operator on \(D\). The author is interested in the construction of function systems \(\{\varphi_ n\}\) which satisfies homogeneous boundary conditions on \(D\) and which are complete simultaneously in \(L_ 2(D)\) and in the energetic space \(H_ P \subseteq L_ 2 (D)\) associated to \(P\). The main applications are \(P\)= Laplacian and \(D\)= interval, rectangle, hollow cylinder. To this aim the author uses a refinement of the Stone-Weierstraß theorem in the algebra \(C(D)\) to weaken the assumption of separating points. Key results are the following: Theorem A: If \(\{\varphi_ n\} = \{\overline \varphi_ n\} \subseteq C(D)\) is a system which ``\(d \mu\)-almost'' separates the points of \(D\), then the algebra generated by \(\{\varphi_ n\}\) is complete in \(L_ p (D)\) for \(1 \leq p < \infty\). Theorem B: If there is some function \(f \in C(D)\) with \(1/f \in C(D)\) and some sequence \(\{\varphi_ n\} \subseteq C(D)\) such that \(\{f \circ P \varphi_ n\}\) is complete in \(C(D)\) and in \(L^ 2(D)\), then \(\{\varphi_ n\}\) is complete in the energetic space \(H_ P\) with the energetic norm \(\| g \|^ 2_ P = (Pg,g)\).