Copyright © 2010 K. Farahmand and T. Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper provides asymptotic estimates for the expected number of real zeros of two different forms of random trigonometric polynomials, where the coefficients of polynomials are normally distributed random variables with different means and variances. For the polynomials in the form of a0+a1cosθ+a2cos2θ+⋯+ancosnθ and a0+a1cosθ+b1sinθ+a2cos2θ+b2sin2θ+⋯+ancosnθ+bnsinnθ, we give a closed form for the above expected value. With some mild assumptions on the coefficients we allow the means and variances of the coefficients to differ from each others. A case of reciprocal random polynomials for both above cases is studied.