An asymptotic expansion for the expected number of real zeros of real random polynomials spanned by OPUC

• Published in: Journal of mathematical analysis and applications Vol. 469; no. 1; pp. 428 - 446
• Main Authors: Aljubran, Hanan, Yattselev, Maxim L.
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.01.2019
• Subjects: Asymptotic expansion; Expected number of real zeros; Orthogonal polynomials on the unit circle; Random polynomials; Expected number of real zeros; Asymptotic expansion; Random polynomials; Orthogonal polynomials on the unit circle
• ISSN: 0022-247X; 1096-0813
• DOI: 10.1016/j.jmaa.2018.09.022

Let {φi}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 En(μ), of random polynomialsPn(z):=∑i=0nηiφi(z), where η0,…,ηn 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 En(|dξ|) admits an asymptotic expansion of the formEn(|dξ|)∼2πlog⁡(n+1)+∑p=0∞Ap(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 En(μ) admits an analogous expansion with the coefficients Ap depending on the measure μ for p≥1 (the leading order term and A0 remain the same).