Real Zeros of Random Cosine Polynomials with Palindromic Blocks of Coefficients

Abstract

It is well known that a random cosine polynomial \({V_n}\left(x \right) = \sum\nolimits_{j = 0}^n {{a_j}\cos \left({jx} \right)} \), x ∈ (0, 2π), with the coefficients being independent and identically distributed (i.i.d.) real-valued standard Gaussian random variables (asymptotically) has \(2n/\sqrt 3 \) expected real roots. On the other hand, out of many ways to construct a dependent random polynomial, one is to force the coefficients to be palindromic. Hence, it makes sense to ask how many real zeros a random cosine polynomial (of degree n) with identically and normally distributed coefficients possesses if the coefficients are sorted in palindromic blocks of a fixed length ℓ In this paper, we show that the asymptotics of the expected number of real roots of such a polynomial is \({{\rm{K}}_\ell} \times 2n/\sqrt 3 \), where the constant Kℓ (depending only on ℓ) is greater than 1, and can be explicitly represented by a double integral formula. That is to say, such polynomials have slightly more expected real zeros compared with the classical case with i.i.d. coefficients.