On completeness of orthogonal systems and Dirac deltas (Q1899954)

Given a positive measure \(\mu\) supported on a set \(\Omega \subseteq \mathbb{C}\), an orthonormal system \(\{ \varphi_n \}_{n\geq 0}\) and a point \(a\in \Omega\), the authors study the relationship among \(\mu( \{a \})\), the kernels \(K_n (a,a)= \sum_{k=1}^n \varphi_k (a) \overline {\varphi_k (a)}\) and the denseness of \(\text{span} \{\varphi_n \}_{n\geq 0}\) in \(L^2 (\mu)\) and in \(L^2 (\nu)\) where \(\nu= \mu+ M\delta_a\). An illustrative example is also considered.