On the number of real zeros of a random trigonometric polynomial

Abstract

Abstract. For the random trigonometric polynomial N 2 s,(/)cosn0, n-l where g"(t), 0 < / < 1, are dependent normal random variables with mean zero, variance one and joint density functionwhere A/-1 is the moment matrix with py = p, 0 < p < 1, i ¥*j, i,j m 1, 2.N and a is the column vector, we estimate the probable number of zeros.1. Consider the random trigonométrie polynomial N (1.1) <t>(9) = $(t, 9) =% gn(t) cos n9, where g"(t), 0 < t < 1, are dependent normal random variables with mean zero, variance one and joint density functionwhere M ~x is the moment matrix with pi} = p, 0 < p < 1, i =£j, i,j = 1, 2, . . . , N, and ä is the column vector whose transpose is 5' = (gx(t),..., gN(t)). In this paper we calculate the probable number of zeros of (1.1). We prove the following.Theorem 1. In the interval 0 < 9 < 2tt all save certain exceptional set of functions <b (9) have (1.3) (2Ar/3'/2) + 0(ATn/,3+") zeros, when N is large. The measure of the exceptional set does not exceed N ~2e>, where ex is any positive number less than 1/13.The particular case when p = 0, that is the case when gn(t) are independent normal random variables, was considered by Dunnage [3] and proved the following.

Zeros of Fekete polynomials

“…It is known that a "random" trigonometric polynomial of degree p has p/V3 zeros in [0,1) (see [7]), so one might guess that KQ = 1/\/3 ^ 0.5773... However this is not the case.…”

Algebraic polynomials with random non-symmetric coefficients

“…This shows they have significantly more zeros than algebraic polynomials. This number remains the same when we pass to the case of non-zero µ, see [7] or [17].…”

The Average Number of Real Zeros of a Random Trigonometric Polynomial

“…For example, Dunnage [4] studied polynomials T n = n k=1 a k cos k with a k s independent identically distributed standard normal random variables and estimated that a probable number of real zeros in the interval 0 ≤ ≤ 2 to be 2 √ 3 n + O n 11/13 log n 3/13 when n is large. Later, Das [3] considered the generalization T n = n k=1 a k b k cos k , where the a k s are also independent identically distributed standard normal random variables with b k = k p for p ≥ −1/2 and found that the expected number of real zeros in 0 ≤ ≤ 2 to be 2p+1 2p+3 1/2 2n + O √ n for large n. Sambandham and Maruthachalam [8] obtained the same result for n large when the a k s are dependent random variables with constant correlation between any two of them. We refer the reader to [1,5] for detail results on random polynomials.…”