Real zeros of random trigonometric polynomials with $\ell$-periodic coefficients
Volume 174 / 2023
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.
