Mean number of real zeros of a random trigonometric polynomial

Abstract

If ax, a2, ... , an are independent, normally distributed random variables with mean 0 and variance 1, and if vn is the mean value of the number of zeros on the interval (0, 2k) of the trigonometric polynomial ax cosx + a2 cos 2x + + an cos nx , then vn = 3"1/2{(2/j + I) + Dx + (2n + l)'1 D2 + (2n + l)'2D}} + 0{(2n + l)"3} , in which Dx = 0.232423 , D2 = -0.25973 , and D3 = 0.2172