Mean number of real zeros of a random trigonometric polynomial. III

[摘要] Ifa1,a2,…,anare independent, normally distributed random variables with mean0and variance1, and ifvnis the mean number of zeros on the interval(0,2π)of the trigonometric polynomiala1cosx+2½a2cos2x+…+n½ancosnx,thenvn=2−½{(2n+1)+D1+(2n+1)−1D2+(2n+1)−2D3}+O{(2n+1)−3}, in whichD1=−0.378124,D2=−12,D3=0.5523. After tabulation of5Dvalues ofvnwhenn=1(1)40, we find that the approximate formula forvn, obtained from the above result when the error term is neglected, produces5Dvalues that are in error by at most10−5whenn≥8, and by only about0.1%whenn=2.

[学科分类] 应用数学