Asymptotic Zero Distribution of Random Polynomials Spanned by General Bases

• Published in: Modern Trends in Constructive Function Theory Vol. 661; pp. 121 - 140
• Main Author: Pritsker, Igor E.
• Format: Book Chapter
• Language: English
• Published: Providence, Rhode Island American Mathematical Society 31.03.2016
• Series: Contemporary Mathematics
• Subjects: expected number of zeros; Polynomials; random coefficients; random orthogonal polynomials; uniform distribution
• ISBN: 1470425343; 9781470425340
• ISSN: 0271-4132; 1098-3627
• DOI: 10.1090/conm/661/13278

Zeros of Kac polynomials spanned by monomials with i.i.d. random coefficients are asymptotically uniformly distributed near the unit circumference. We give estimates of the expected discrepancy between the zero counting measure and the normalized arclength on the unit circle. Similar results are established for polynomials with random coefficients spanned by different bases, e.g., by orthogonal polynomials. We show almost sure convergence of the zero counting measures to the corresponding equilibrium measures, and quantify this convergence, relying on the potential theoretic methods developed for deterministic polynomials. Applications include estimates of the expected number of zeros in various sets. Random coefficients may be dependent and need not have identical distributions in our results.