Mean number of real zeros of a random trigonometric polynomial IV. (1997) - Wilkins, J. Ernest
Abstract : If a j ( j = 1, 2, …, n ) are independent, normally distributed random variables with mean 0 and variance 1, if p is one half of any odd positive integer except one, and if v n p is the mean number of zeros on ( 0, 2 π ) of the trigonometric polynomial a 1 cos x + 2 p a 2 cos 2 x + … + n p a n cos n x, then v n p = μ p ( 2 n + 1 ) + D 1 p + ( 2 n + 1 ) − 1 D 2 p + ( 2 n + 1 ) − 2 D 3 p + O ( 2 n + 1 ) − 3 , in which μ p = ( 2 p + 1 ) / ( 2 p + 3 ) ½, and D 1 p, D 2 p and D 3 p are explicitly stated constants.