How Many Number of Real Zeros of A Random Trigonometric Polynomial
Keywords:
Independent, identically distributed random variables, random random algebraic polynomialalgebraic polynomial,, random algebraic equation, real roots, domain of attraction of the normal law, slowly varying functionAbstract
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 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. 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.
