On uniform boundedness and uniform asymptotics for orthogonal polynomials on the unit circle (Q1270851)

This paper is concerned with two properties of sequences of orthonormal polynomials \(\{\varphi_n\}\) on the unit circle 1. Uniform asymptotic representation: \(\varphi_n(e^{z})=e^{nz}S^{-1}(z)(1+\varepsilon_n(z))\) with \(z=i\vartheta\), \(\varepsilon_n = o(1), n\rightarrow\infty\), uniformly in \(\vartheta\) and \[ S(w)=\exp\left\{{1\over 4\pi}\int_0^{2\pi} {e^{i\vartheta}+w \over e^{i\vartheta}-w}\log\mu'd\vartheta\right\} \] is the Szegő function for the positive Borel measure \(\mu\) with infinite support on \([0,2\pi)\). 2. Uniform boundedness: \(\varphi_n(e^{i\vartheta})=O(1)\), \(n\rightarrow\infty\). In the literature there are several conditions known: A. Zygmund's condition for 1: \(\{\varphi_n\}\) orthonormal w.r.t. a positive weight function \(f\in C\) that satisfies \(\sum_{n=1}^{\infty} {\omega(1/n,f)\over n} <\infty\). B. Baxter's theorem for 1: The Fourier series for the positive weight function \(f\) converges absolutely. C. Ya. L. Geronimus' condition for 2: \(0<m<f(\vartheta)\leq M\) a.e. and \(f\in\text{ Lip}(2,1/2)\). The authors show by three straightforward examples that the conditions A, B, and C are not necessary.