Turán measures
In 1988 \textit{M. Alfaro} and \textit{L. Vigil} [J. Approximation Theory 53, 195-197 (1988; Zbl 0641.33014)] proved that there is a probability measure \(\mu\) on the unit circle with the set of zeros (the set of zeros of all orthogonal with respect to \(\mu\) polynomials) dense in the unit disk (Turán's conjecture). In the present paper the author shows that the set of such measures (called Turán measures) is rather massive. Theorem 1. Given a measure \(\mu\) with parameters \(\{a_n\}\) and an arbitrary infinite set \(\Lambda\subset\mathbb N\) there exists a Turán measure \(\mu'\) with parameters \(\{a_n'\}\) satisfying \(a_n=a_n'\) for \(n\notin\Lambda\). In particular, there are Turán measures with parameters \(a_n\in\ell^p\) for all \(p>0\) as well as Turán measures with infinitely differentiable densities.
scientific article; zbMATH DE number 1924734