Real zeros of random trigonometric polynomials with pairwise equal\n blocks of coefficients

2019/05/30 by Ali Pirhadi, Pirhadi, Ali
Mathematics · #26C10 #30C15 #Analytic and geometric function theory #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Geometry and complex manifolds #Mathematical functions and polynomials #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.1905.13349

openalex publication_date 2019/05/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

It is well known that the expected number of real zeros of a random cosine\npolynomial V_n(x) = \∑_ j=0 ⁿ a_j \cos (j x) , x \∈ (0,2\π) ,\nwith the a_j being standard Gaussian i.i.d. random variables is\nasymptotically 2n / \√(3) . On the other hand, some of the previous works\non the random cosine polynomials with dependent coefficients show that such\npolynomials have at least 2n / \√(3) expected real zeros lying in one\nperiod. In this paper we investigate two classes of random cosine polynomials\nwith pairwise equal blocks of coefficients. First, we prove that a random\ncosine polynomial with the blocks of coefficients being of a fixed length and\nsatisfying A_2j=A_2j+1 possesses the same expected real zeros as the\nclassical case. Afterwards, we study a case containing only two equal blocks of\ncoefficients, and show that in this case significantly more real zeros should\nbe expected compared to those of the classical case.\n

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