On the zeros of asymptotically external polynomial sequences in the plane

JOURNAL OF APPROXIMATION THEORY

卷:191

Article

Saff, E. B.¹  Stylianopoulos, N.² 

[1] Vanderbilt Univ, Ctr Construct Approximat, Dept Math, Nashville, TN 37240 USA

[2] Univ Cyprus, Dept Math & Stat, CY-1678 Nicosia, Cyprus

关键词: Orthogonal polynomials;    Equilibrium measure;    External polynomials;    Zeros of polynomials;   

DOI  :  10.1016/j.jat.2014.10.003

来源: Elsevier

【 摘 要 】

Let E be a compact set of positive logarithmic capacity in the complex plane and let {P-n (z)}(1)T-infinity be a sequence of asymptotically extremal monic polynomials for E in the sense that lim(n ->infinity) sup parallel to P-n parallel to(1/n)(E) <= cap(E). The purpose of this note is to provide sufficient geometric conditions on E under which the (full) sequence of normalized counting measures of the zeros of {P-n} converges in the weak-star topology to the equilibrium measure on E, as n -> infinity. Utilizing an argument of Gardiner and Pommerenke dealing with the balayage of measures, we show that this is true, for example, if the interior of the polynomial convex hull of E has a single component and the boundary of this component has an inward corner (more generally, a non-convex singularity). This simple fact has thus far not been sufficiently emphasized in the literature. As applications we mention improvements of some known results on the distribution of zeros of some special polynomial sequences. (C) 2014 Elsevier Inc. All rights reserved.

【 授权许可】

Free