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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).
Computational Methods and Function Theory – Springer Journals
Published: May 21, 2019
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