AVERAGE NUMBER OF REAL ZEROS OF RANDOM TRIGONOMETRIC POLYNOMIAL

Abstract

Let EN( T; Φ’ , Φ’’ ) denote the average number of real zeros
of the random trigonometric polynomial
T=Tn( Φ, ω )=
a bK
k
n
K
K
cos
1

In the interval (Φ’ , Φ’’ ). Assuming that ak(ω ) are independent random
variables identically distributed according to the normal law and that bk = kp
(p ≥ 0) are positive constants, we show that
EN( T : 0, 2π ) ~
n O n
p
p
n
(1 ) 2 log
2 3
2 1
2
1
2








  








Outside an exceptional set of measure at most (2/ n ) where
  
2
2
2
'(log )
4 2 1 2 3
SS n
p p
n
 
 

β = constant S ~ 1 S’ ~ 1
1991 Mathematics subject classification (amer. Math. Soc.): 60 B 99