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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 Theory – Springer Journals

Published: May 21, 2019

References

  • Discrepancy of Signed Measures and Polynomial Approximation
    Andrievskii, VV; Blatt, H-P
  • Lectures on Complex Approximation
    Gaier, D
  • On the zeros of sequences of polynomials
    Grothmann, R
  • Some Random Series of Functions
    Kahane, JP
  • Comparing norms of polynomials in one and several variables
    Pritsker, I
  • Equidistribution of zeros of random polynomials
    Pritsker, I; Ramachandran, K
  • Functions of a Complex Variable: Constructive Theory
    Smirnov, VI; Lebedev, NA
  • Strong asymptotics of Bergman polynomials over domains with corners and applications
    Stylianopoulos, N
  • Polynomials Orthogonal Over a Region and Bieberbach Polynomials
    Suetin, PK