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- W2990204699 abstract "Abstract Let { φ i } i = 0 ∞ be a sequence of orthonormal polynomials on the unit circle with respect to a positive Borel measure μ that is symmetric with respect to conjugation. We study asymptotic behavior of the expected number of real zeros, say E n ( μ ) , of random polynomials P n ( z ) : = ∑ i = 0 n η i φ i ( z ) , where η 0 , … , η n are i.i.d. standard Gaussian random variables. When μ is the acrlength measure such polynomials are called Kac polynomials and it was shown by Wilkins that E n ( | d ξ | ) admits an asymptotic expansion of the form E n ( | d ξ | ) ∼ 2 π log ( n + 1 ) + ∑ p = 0 ∞ A p ( n + 1 ) − p (Kac himself obtained the leading term of this expansion). In this work we generalize the result of Wilkins to the case where μ is absolutely continuous with respect to arclength measure and its Radon–Nikodym derivative extends to a holomorphic non-vanishing function in some neighborhood of the unit circle. In this case E n ( μ ) admits an analogous expansion with the coefficients A p depending on the measure μ for p ≥ 1 (the leading order term and A 0 remain the same)." @default.
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- W2990204699 date "2019-01-01" @default.
- W2990204699 modified "2023-10-18" @default.
- W2990204699 title "An asymptotic expansion for the expected number of real zeros of real random polynomials spanned by OPUC" @default.
- W2990204699 hasPublicationYear "2019" @default.
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