An asymptotic expansion for the expected number of real zeros of real random polynomials spanned by OPUC |
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Authors: | Hanan Aljubran Maxim L. Yattselev |
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Affiliation: | Department of Mathematical Sciences, Indiana University-Purdue University Indianapolis, 402 North Blackford Street, Indianapolis, IN 46202, USA |
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Abstract: | Let 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 , of random polynomials where 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 admits an asymptotic expansion of the form (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 admits an analogous expansion with the coefficients depending on the measure μ for (the leading order term and remain the same). |
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Keywords: | Random polynomials Orthogonal polynomials on the unit circle Expected number of real zeros Asymptotic expansion |
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