Mean number of real zeros of a random trigonometric polynomial. III. (1995) - Wilkins, J. Ernest; Souter, Shantay A.
Abstract : If a 1, a 2, …, a n are independent, normally distributed random variables with mean 0 and variance 1, and if v n is the mean number of zeros on the interval ( 0, 2 π ) of the trigonometric polynomial a 1 cos x + 2 ½ a 2 cos 2 x + … + n ½ a n cos n x, then v n = 2 − ½ { ( 2 n + 1 ) + D 1 + ( 2 n + 1 ) − 1 D 2 + ( 2 n + 1 ) − 2 D 3 } + O { ( 2 n + 1 ) − 3 }, in which D 1 = − 0.378124, D 2 = − 1 2, D 3 = 0.5523 . After tabulation of 5 D values of v n when n = 1 ( 1 ) 40, we find that the approximate formula for v n, obtained from the above result when the error term is neglected, produces 5 D values that are in error by at most 10 − 5 when n ≥ 8, and by only about 0.1 % when n = 2 .