Zeros of random trigonometric polynomials with dependent coefficients

Abstract

It is well known that the expected number of real zeros of a random cosine polynomial (of degree $ n $)\begin{equation*} V_n(x) = \sum_ {j=0} ^{n} a_j \cos (j x) , \quad x \in (0,2\pi) , \end{equation*} where the coefficients $ a_j $ are independent and identically distributed (i.i.d.) real-valued standard Gaussian random variables, is asymptotically $ 2n / \sqrt{3} $. To the best of our knowledge, the above asymptotic relation has always been the lower bound for the expected number of real zeros of $ V_n $ when the $ a_j $ employ different settings. However, this inequality is sharp for most of the cases that have been considered so far. Moreover, out of various ways to establish a set of dependent coefficients, one is to sort out the coefficients in the blocks of the same length, and then identify certain blocks. As one may expect, the expected number of real zeros of these polynomials is subject to the way we identify the blocks, yet it might be independent of the size of the blocks. In this manuscript, we investigate four cases of random cosine polynomials where the blocks of the coefficients are identified in different fashions. The cases we study include the adjacent, palindromic, and periodic blocks as well as the case involving only two blocks, each of which possesses a different expected number of real zeros from one another.