A density problem for orthogonal rational functions (Q1301956)

A sequence of points in the open unit disk in the complex plane and the relations \(\mathbb{B}_0=1\) and \(\mathbb{B}_n(z)= \prod^n_{i=0} {\overline {\alpha_k} \over|\alpha_k|} {\alpha_k- z\over 1-\overline {\alpha_k} z}\), \(n=1,2,\dots\), were considered. The moment problem involves finding a function \(\mu\) on \([-\pi,\pi]\) such that \[ \langle f,g\rangle= \int^\pi_{- \pi} f(e^{i\theta}) \overline{g (e^{i\theta})} d\mu(\theta) \quad \text{for} \quad f,g\in {\mathcal L}. \] A necessary and sufficient condition \((N\)-extremal) was given on a solution of the moment problem in order that \({\mathcal L}\) is dense in \(L^2\). The results are based on nested disks theories related to the corresponding moment problem.