On exceptional sets of asymptotic relations for general orthogonal polynomials (Q1899358)

The author studies the \(n\)th root asymptotic behavior of the orthonormal polynomials \(q_n(z)\), corresponding to an arbitrary measure in the plane, in terms of the Green function \(g_\Omega(z)\) for the outer domain \(\Omega\) of the support of \(\mu\). He succeeds in improving upon the most general results to date [cf. \textit{H. Stahl} and \textit{V. Totik}, General orthogonal polynomials and its applications (1992; Zbl 0791.33009)]. The main results are: A. \(\limsup_{n\to \infty} |q_n(z)|^{1/n}|\geq e^{g_\Omega(z)}\) everywhere in \(\Omega\), B. \(\limsup_{n\to \infty} |q(z)|^{1/n}|\geq 1\) everywhere on \(\partial\Omega\), outside the discrete spectrum of \(\mu\). Further extensions given are: \(\text{A}'\). The convergence in A is locally uniformly in \(z\) for an infinite subsequence of natural numbers (depending on the center of the neighborhood), \(\text{B}'\) \((\sum^\infty_{n= 0} |q^2_n(z)|)^{- 1}= \mu(\{z\})\) everywhere on \(\partial\Omega\). Finally, it is shown that the exceptional set in B cannot be reduced and an application to convergence of diagonal Padé approximants of Markov-functions is given. An important step in the study of asymptotics of general orthogonal polynomials.