Expected number of real roots of random trigonometric polynomials
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}