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Globalization of real zeros of a random trigonometric polynomial

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Author: 
Dr. P. K. Mishra and Dipty Rani Dhal
Abstract: 

Let EN(T; Φ’, Φ’’) denote the average number of real roots of the random trigonometric polynomial
T=Tn(θ, ω)=
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 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.

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