ON A RESULT CONCERNING REAL ZEROS OF A RANDOM TRIGONOMETRIC POLYNOMIAL

Abstract

Let EN( T; Φ’ , Φ’’ ) denote the average number of real
roots of the random trigonometric polynomial
T=Tn( θ, ω )=
a  k
n
K
K
cos
1

In the interval (Φ’ , Φ’’ ). Clearly , T can have at most 2n zeros in the
interval ( 0, 2π ) .Assuming that ak(ω )s to be mutually independent
identically distributed normal random variables , Dunnage has shown that
in the interval 0 ≤ θ ≤ 2π all save a certain exceptional set of the
functions (Tn ( θω )) have
  




  13
3
13
11
log
3
2
O n n
n
zeros when n is
large. We consider the same family of trigonometric polynomials and use
the Kac_rice formula for the expectation of the number of real roots
and obtain that
EN ( T ; 0 , 2π ) ~ (log )
6
2
O n
n

This result is better than that of Dunnage since our constant is (1/√2)
Times his constant and our error term is smaller . the proof is based
on the convergence of an integral of which an asymptotic estimation is
obtained .
1991 Mathematics subject classification (amer. Math. Soc.): 60 B 99.