On second kind measures and polynomials on the unit circle
Notations and background results. Let \(d\sigma\) be a positive Borel measure with infinite support. \(d\sigma(t)= \phi(t)dt+ d\sigma_ s(t)\) is its Lebesgue decomposition, where \(\phi\in L^ 1[- \pi,\pi]\) is a density function. The moment sequence is \(c_ k(d\sigma)= {1\over 2\pi} \int^ \pi_{-\pi} \zeta^ k d\sigma\), \(k= 0,1,\dots\) and the unique system of associated orthonormal polynomials \(\{\phi_ 0\}\) satisfy the relations \[ {1\over 2\pi} \int^ \pi_{-\pi} \phi_ n(\zeta) \overline{\phi_ m(\zeta)} d\sigma(t)= \delta_{nm},\quad \zeta= e^{it}. \] The Carathéodory function (C-function) \(F(z)\) is defined by \[ F(z)= {1\over 2\pi c_ 0} \int^ \pi_{-\pi} S(t, z) d\sigma,\quad S(t, z)= {e^{it}+ z\over e^{it}- z} \] satisfies \({\mathfrak R} F(z)> 0\), \(| z|< 1\), \(\lim{\mathfrak R}_{r\to 1- 0} F(re^{i\theta})= {\phi(\theta)\over c_ 0}\) a.e. on \([- \pi,\pi]\). \(d\sigma\) is said to be in Szegö class if \(\int^ \pi_{-\pi} \ln \sigma'(t) dt>- \infty\). In that case the principal tool is the function \[ D(\sigma, z)= \exp\Biggl\{{1\over 4\pi} \int^ \pi_{- \pi} {\zeta+ z\over \zeta- z}\ln \sigma'(t) dt\Biggr\}. \] For example \(| D(\sigma, e^{it})|^ 2\overset{\text{a.e.}} =\sigma'(t)\) and \(D(\sigma)\in H^ 2(| z|< 1)\). The author studies the so-called second kind polynomials and measures defined by \[ \psi_ 0(z)= {1\over \sqrt{c_ 0}},\;\psi_ n(z)= {1\over 2\pi c_ 0} \int^ \pi_{-\pi} S(t,z)[\phi_ n(e^{it})- \phi_ n(z)] d\sigma(t),\quad n= 1,2,\dots\;. \] The \(\psi_ n\) are orthogonal with respect to a uniquely determined second kind measure \(d\tau(t)\). The corresponding C-function \[ G(z)= {1\over 2\pi c_ 0} \int^ \pi_{- \pi} S(t, z) d\tau(t)= F(z)^{- 1}. \] For a polynomial \(P(z)\) the conjugate transform is \(P^*(z)= z^ n\overline{P({1\over \overline z})}\). The main result of the paper is the absolute continuity of the second kind measure \(d\tau\) on any interval \([a, b]\subset [- \pi,\pi]\) with respect to the first kind measure \(d\sigma\) provided \(\phi^{-1}\in L^ p[a, b]\) for some \(p> 1\). As an example, the author introduces the second kind measure \(d\tau\) corresponding to the generalized Jacobi weight functions on the unit circle \(w(t)\) defined by \(w(t)= h(t) \prod^ N_{\nu= 1} |\zeta- \zeta_ \nu|^{2\gamma_ \nu}\), \(\zeta_ \nu= e^{it_ \nu}\), \(t_ \nu\in [- \pi,\pi]\) distinct, \(\gamma_ \nu> -{1\over 2}\), where one assumes \(h\in C_{2\pi}\) (continuous, periodic), \(h> 0\), \(w(t, h)t^{- 1}\in L^ 1(0, \pi)\). Finally, the author studies the asymptotic formula \[ \lim_{n\to\infty} \psi^*_ n(e^{i\theta})= D^{- 1}(e^{i\theta}, \tau) \] in case \(d\sigma\) belongs to the Szegö class and some additional conditions.
scientific article; zbMATH DE number 756632