Expected number of real roots of random trigonometric polynomials

TL;DR

This paper derives the asymptotic expected number of real roots of certain random trigonometric polynomials, showing it scales linearly with degree and depends on the interval and coefficient distribution.

Contribution

It provides a precise asymptotic formula for the expected number of real roots of random trigonometric polynomials with i.i.d. coefficients, extending understanding of their root distribution.

Findings

Expected roots scale linearly with polynomial degree n.

Asymptotic expectation depends on interval length and coefficient distribution.

Explicit formula involving exponential decay with respect to u^2/2.

Abstract

We investigate the asymptotics of the expected number of real roots of random trigonometric polynomials $$ X_n(t)=u+\frac{1}{\sqrt{n}}\sum_{k=1}^n (A_k\cos(kt)+B_k\sin(kt)), \quad t\in [0,2\pi],\quad u\in\mathbb{R} $$ whose coefficients $A_k, B_k$, $k\in\mathbb{N}$, are independent identically distributed random variables with zero mean and unit variance. If $N_n[a, b]$ denotes the number of real roots of $X_n$ in an interval $[a,b]\subseteq [0,2\pi]$, we prove that $$ \lim_{n\rightarrow\infty} \frac{\mathbb{E} N_n[a,b]}{n}=\frac{b-a}{\pi\sqrt{3}} e^{-\frac{u^2}{2}}. $$