Instytut Matematyczny Polskiej Akademii Nauk / Institute of Mathematics / Publishing house / Journals and Serials / Colloquium Mathematicum / All issues

Abstract

The large degree asymptotic of the expected number of real zeros of a random trigonometric polynomial $$T_n(x) = \sum_{j=0} ^{n} (a_j \cos (j x) + b_j \sin (j x)), \quad x \in (0,2\pi ),$$ with i.i.d. real-valued standard Gaussian coefficients is known to be $2n / \sqrt {3}$. In this article, we consider quite a different and extreme setting on the set of coefficients of $T_n$. We show that a random trigonometric polynomial of degree $n$ with $\ell$-periodic i.i.d. Gaussian coefficients is expected to have significantly more real zeros compared to the classical case with i.i.d. Gaussian coefficients. More precisely, the expected number of real zeros of $T_n$ is proportional to the degree $n$ with a proportionality constant $\mathrm C_{\ell ,r} \in (\sqrt{2},2]$ where $\ell$ is the period of the coefficients, and $r$ is the remainder of the Euclidean division of $n$ by $\ell$. This constant is explicitly represented by a double integral formula. The case $r=0$ is marked as a special one since $T_n$ in such a case asymptotically obtains the largest possible number of real zeros.