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August 2020 Real zeros of random trigonometric polynomials with pairwise equal blocks of coefficients
Ali Pirhadi
Rocky Mountain J. Math. 50(4): 1451-1471 (August 2020). DOI: 10.1216/rmj.2020.50.1451


It is well known that the expected number of real zeros of a random cosine polynomial Vn(x)=j=0naj cos(jx), x(0,2π), with the aj being standard Gaussian i.i.d. random variables, is asymptotically 2n3. On the other hand, some of the previous works on the random cosine polynomials with dependent coefficients show that such polynomials have at least 2n3 expected real zeros lying in one period. We investigate two classes of random cosine polynomials with pairwise equal blocks of coefficients. First, we prove that a random cosine polynomial with the blocks of coefficients being of a fixed length and satisfying A2j+1=A2j possesses the same expected real zeros as the classical case. Afterwards, we study a case containing only two equal blocks of coefficients, and show that in this case significantly more real zeros should be expected compared with those of the classical case.


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Ali Pirhadi. "Real zeros of random trigonometric polynomials with pairwise equal blocks of coefficients." Rocky Mountain J. Math. 50 (4) 1451 - 1471, August 2020.


Received: 2 June 2019; Revised: 13 January 2020; Accepted: 17 January 2020; Published: August 2020
First available in Project Euclid: 29 September 2020

zbMATH: 07261874
MathSciNet: MR4154817
Digital Object Identifier: 10.1216/rmj.2020.50.1451

Primary: 30C15
Secondary: 26C10

Keywords: dependent coefficients , expected number of real zeros , random trigonometric polynomials

Rights: Copyright © 2020 Rocky Mountain Mathematics Consortium


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Vol.50 • No. 4 • August 2020
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