Mean number of real zeros of a random trigonometric polynomial

Abstract:

If ${a_1},{a_2}, \ldots ,{a_n}$ are independent, normally distributed random variables with mean 0 and variance 1, and if ${\nu _n}$ is the mean value of the number of zeros on the interval $(0,2\pi )$ of the trigonometric polynomial ${a_1}\cos x + {a_2}\cos 2x + \cdots + {a_n}\cos nx$, then \[ {\nu _n} = {3^{1/2}}\{ (2n + 1) + {D_1} + {(2n + 1)^{ - 1}}{D_2} + {(2n + 1)^{ - 2}}{D_3}\} + O\{ {(2n + 1)^{ - 3}}\} ,\] in which ${D_1} = 0.232423 \cdots ,{D_2} = - 0.25973 \cdots$, and ${D_3} = 0.2172 \cdots$.