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Natural Boundary and Zero Distribution of Random Polynomials in Smooth Domains

Natural Boundary and Zero Distribution of Random Polynomials in Smooth Domains We consider the zero distribution of random polynomials of the form $$P_n(z) = \sum _{k=0}^n a_k B_k(z)$$ P n ( z ) = ∑ k = 0 n a k B k ( z ) , where $$\{a_k\}_{k=0}^{\infty }$$ { a k } k = 0 ∞ are non-trivial i.i.d. complex random variables with mean 0 and finite variance. Polynomials $$\{B_k\}_{k=0}^{\infty }$$ { B k } k = 0 ∞ are selected from a standard basis such as Szegő, Bergman, or Faber polynomials associated with a Jordan domain G whose boundary is $$C^{2, \alpha }$$ C 2 , α smooth. We show that the zero counting measures of $$P_n$$ P n converge almost surely to the equilibrium measure on the boundary of G. We also show that if $$\{a_k\}_{k=0}^{\infty }$$ { a k } k = 0 ∞ are i.i.d. random variables, and the domain G has analytic boundary, then for a random series of the form $$f(z) =\sum _{k=0}^{\infty }a_k B_k(z),$$ f ( z ) = ∑ k = 0 ∞ a k B k ( z ) , $$\partial {G}$$ ∂ G is almost surely the natural boundary for f(z). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Computational Methods and Function Theory Springer Journals

Natural Boundary and Zero Distribution of Random Polynomials in Smooth Domains

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Publisher
Springer Journals
Copyright
Copyright © 2019 by Springer-Verlag GmbH Germany, part of Springer Nature
Subject
Mathematics; Analysis; Computational Mathematics and Numerical Analysis; Functions of a Complex Variable
ISSN
1617-9447
eISSN
2195-3724
DOI
10.1007/s40315-019-00273-0
Publisher site
See Article on Publisher Site

Abstract

We consider the zero distribution of random polynomials of the form $$P_n(z) = \sum _{k=0}^n a_k B_k(z)$$ P n ( z ) = ∑ k = 0 n a k B k ( z ) , where $$\{a_k\}_{k=0}^{\infty }$$ { a k } k = 0 ∞ are non-trivial i.i.d. complex random variables with mean 0 and finite variance. Polynomials $$\{B_k\}_{k=0}^{\infty }$$ { B k } k = 0 ∞ are selected from a standard basis such as Szegő, Bergman, or Faber polynomials associated with a Jordan domain G whose boundary is $$C^{2, \alpha }$$ C 2 , α smooth. We show that the zero counting measures of $$P_n$$ P n converge almost surely to the equilibrium measure on the boundary of G. We also show that if $$\{a_k\}_{k=0}^{\infty }$$ { a k } k = 0 ∞ are i.i.d. random variables, and the domain G has analytic boundary, then for a random series of the form $$f(z) =\sum _{k=0}^{\infty }a_k B_k(z),$$ f ( z ) = ∑ k = 0 ∞ a k B k ( z ) , $$\partial {G}$$ ∂ G is almost surely the natural boundary for f(z).

Journal

Computational Methods and Function TheorySpringer Journals

Published: May 21, 2019

References