ON A RESULT CONCERNING REAL ZEROS OF A RANDOM TRIGONOMETRIC POLYNOMIAL - MISHRA, DR. P. K.; DHAL, DIPTY RANI (IJRDO -JOURNAL OF MATHEMATICS, 2016)
Let EN( T; Φ’ , Φ’’ ) denote the average number of realroots of the random trigonometric polynomialT=Tn( θ, ω )=a  knKKcos1In the interval (Φ’ , Φ’’ ). Clearly , T can have at most 2n zeros in theinterval ( 0, 2π ) .Assuming that ak(ω )s to be mutually independentidentically distributed normal random variables , Dunnage has shown thatin the interval 0 ≤ θ ≤ 2π all save a certain exceptional set of thefunctions (Tn ( θω )) have    1331311log32O n nnzeros when n islarge. We consider the same family of trigonometric polynomials and usethe Kac_rice formula for the expectation of the number of real rootsand obtain thatEN ( T ; 0 , 2π ) ~ (log )62O nn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 basedon the convergence of an integral of which an asymptotic estimation isobtained .1991 Mathematics subject classification (amer. Math. Soc.): 60 B 99.