Moment problem for rational orthogonal functions on the unit circle (Q1364291)

Let \((z_n)\) and \((a_n)\) be sequences of complex numbers in the open unit disk with \(\sum^\infty_{n=1}(1-|z_n|)= \infty\). The author shows that if a sequence of rational functions \(\phi_n\) is defined by a recurrence relation of the form \[ \psi_n(z)= {z-z_{n-1}\over1-\overline z_nz} \phi_{n- 1}(z)+ a_n{1-\overline z_{n-1}z\over 1-\overline z_nz} \phi^*_{n- 1}(z),\;\phi_n(z)={\psi_n(z)\over \psi^*_n(z_n)}, \] where \(*\) is the sometimes called superstar transformation, and if \(\phi_n\) is normalized appropriately, then there exists a finite positive Borel measure \(\mu\) on the unit circle such that \((\phi_n)\) is orthonormal with respect to \(\mu\). As a consequence, this ensures the existence of measures \(\mu\) such that the zeros of the corresponding orthogonal rational functions \(\phi_n\) are dense in the unit disk.