Different roots a random trigonometric polynomial

Author: 
Mishra P. K and Manasingh A.K

Let EN( T; Φ’ , Φ’’ ) denote the average number of real roots of the random trigonometric polynomial
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 [1] has shown that in the interval 0 ≤ θ ≤ 2π all save a certain exceptional set of the functions (Tn ( θ,ω )) have number of 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) ~
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.

Download PDF: