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Computational Methods and Function Theory

, Volume 19, Issue 3, pp 401–410 | Cite as

Natural Boundary and Zero Distribution of Random Polynomials in Smooth Domains

  • Igor Pritsker
  • Koushik RamachandranEmail author
Article
  • 15 Downloads

Abstract

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

Keywords

Random polynomials Orthogonal polynomials Zero distribution Natural boundary 

Mathematics Subject Classification

MSC 60F05 31A15 30B20 30B30 

Notes

Acknowledgements

Research of the first author was partially supported by the National Security Agency (Grant H98230-15-1-0229) and by the American Institute of Mathematics. We are grateful to the referee for pointing out a correction in the statement of Theorem 1.4, and for other remarks and suggestions which helped in improving the exposition.

References

  1. 1.
    Andrievskii, V.V., Blatt, H.-P.: Discrepancy of Signed Measures and Polynomial Approximation. Springer, New York (2002)CrossRefzbMATHGoogle Scholar
  2. 2.
    Arnold, L.: Zur Konvergenz und Nichtfortsetzbarkeit zufa̋lliger Potenzreihen, pp. 223–234. Statistical Decision Functions, Random Processes, Academia, Prague, Trans. Fourth Prague Conf. on Information Theory (1965)Google Scholar
  3. 3.
    Gaier, D.: Lectures on Complex Approximation. Birkhuser, Boston (1987)CrossRefzbMATHGoogle Scholar
  4. 4.
    Grothmann, R.: On the zeros of sequences of polynomials. J. Approx. Theory 61, 351–359 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Hough, J.B., Krishnapur, M., Peres, Y., Virag, B.: Zeros of Gaussian Analytic Functions and Determinantal Point Processes, University Lecture Series, Vol \(51,\) American Mathematical Society, (2009)Google Scholar
  6. 6.
    Kahane, J.P.: Some Random Series of Functions. Cambridge University Press, Cambridge (1985)zbMATHGoogle Scholar
  7. 7.
    Pritsker, I.: Comparing norms of polynomials in one and several variables. J. Math. Anal. Appl. 216, 685–695 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Pritsker, I., Ramachandran, K.: Equidistribution of zeros of random polynomials. J. Approx. Theory 215, 106–117 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Smirnov, V.I., Lebedev, N.A.: Functions of a Complex Variable: Constructive Theory. MIT Press, Cambridge (1968)zbMATHGoogle Scholar
  10. 10.
    Stylianopoulos, N.: Strong asymptotics of Bergman polynomials over domains with corners and applications. Constr. Approx. 38(1), 59–100 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Suetin, P.K.: Polynomials Orthogonal Over a Region and Bieberbach Polynomials. American Mathematical Society, Providence (1974)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsOklahoma State UniversityStillwaterUSA
  2. 2.Centre for Applicable MathematicsTata Institute of Fundamental ResearchBengaluruIndia

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