Orthogonal polynomials and their zeros. (English) Zbl 0691.42020

P. Turán raised the problem if the zeros of orthogonal polynomials with weight measures on the unit circle can be dense in the unit disk. This problem was solved by M. P. Alfaro and L. Vigil [J. Approximation Theory 53, No.2, 195-197 (1988; Zbl 0641.33014)]. In the paper a very simple solution is given, namely it is shown that the recursion for generating orthogonal polynomials can be done in such a way that we prescribe 1-1 zeros of the different polynomials. For completeness we mention that a recent result of H. Stahl and V. Totik shows that the answer for Turán’s problem is positive in a very general setting: If S is any compact subset of the complex plane then there is a measure on S such that the zeros of the corresponding orthogonal polynomials is dense in the convex hull of S.
Besides Turán’s problem, we also discuss the relation between the recursion coefficients and the zeros. It is also shown that the zeros stay away from the boundary if and only if the weight is analytic. Several other related results are discussed.
Reviewer: P.Nevai


42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)


Zbl 0641.33014