Transformation of polynomials orthogonal on the unit circle (Q1270875)

Let \(\sigma(\phi)\) be a positive Borel measure on the interval \([0,2\pi]\) with infinite support. Then for every \(n\in\mathbb N_0\) there exists a uniquely determined monic orthogonal polynomial \(P_n(z)\) of degree \(n\) which satisfies \[ \int_0^{2\pi}e^{-ij\phi}P_n(e^{i\phi})d\sigma(\phi)=0, \quad j=0,\ldots,n-1. \] These monic orthogonal polynomials can be generated by the recurrence relation \[ P_{n+1}(z)=zP_n(z)-\overline{a_n}P_n^*(z),\quad n\in\mathbb N_0, \] where \(P_n^*(z)=z^n\overline{P_n(1/\overline{z})}\) is the reversed polynomial of \(P_n(z)\). The complex numbers \(a_n=-\overline{P_{n+1}(0)}\) are called reflection coefficients or Schur parameters. Moreover, there is a one-to-one correlation between the measure \(\sigma\) and the Carathéodory function \[ F(z)=\int_0^{2\pi}\frac{e^{i\phi}+z}{e^{i\phi}-z}d\sigma(\phi), \quad | z| <1. \] Let \(\{a_n\}\) be an arbitrary sequence of real numbers with \(| a_n| <1\) for \(n\in\mathbb N_0\), and let \(b_1,\ldots,b_{N-1}\) with \(b_{N-j}=-\overline{b_j}\) be an arbitrary symmetric sequence of \(N-1\) complex numbers with \(| b_j| <1\). The authors give a description of the orthogonal polynomials, the measures, and the Carathéodory functions associated with the reflection coefficients (or Schur parameters) of the form \(\{b_1,\ldots,b_{n-1},a_0,b_1,\ldots,b_{N-1},a_1,\ldots\}\). In fact, they show that the orthogonal polynomials generated by such a sequence of reflection coefficients can be obtained by a transformation of polynomials orthogonal on the unit circle. The measure is of the form \((w(\phi)/\sqrt(\rho(\phi))\theta'(\phi))\) \(d\sigma(\theta(\phi))\), where \(\sigma\) is the orthogonality measure associated with \(\{a_n\}\) and \(w\), \(\rho\) and \(\theta\) are trigonometric polynomials which can be constructed explicitly with the help of \(b_1,\ldots,b_{N-1}\). In particular, \(\theta\) is a function which maps \(N\) subintervals of \([0,2\pi]\) onto the whole interval \([0,2\pi]\). Finally, it is shown that polynomials which are orthogonal with respect to a measure of the form \((w(\phi)/\sqrt(\rho(\phi))\theta'(\phi))d\sigma(\theta(\phi))\), with \(\sigma\) an arbitrary symmetric measure on \([0,2\pi]\) and \(\theta\) a function which maps \(N\) subintervals of \([0,2\pi]\) onto the whole interval \([0,2\pi]\), have reflection coefficients of the above form.