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Real zeros of random trigonometric polynomials with pairwise equal blocks of coefficients. (English) Zbl 1461.30027
The paper deals with a random trigonometric polynomial \(V_n=\sum_{j=0}^{\infty} a_j \cos(jx),\) \(x\in (0, 2\pi).\) Classical results state that if the coefficients \(a_j\) are standard Gaussian i.i.d. random variables, then the expected value \(\mathbb{E}[N_n(0, 2\pi)] \sim \frac{2n}{\sqrt{3}}\) as \(n\to \infty,\) where \(N_n(0, 2\pi)\) denotes the number of real zeros of \(V_n\) in \((0, 2\pi).\) The main focus of the paper is the question on how many real zeros, compared with the case of independent coefficients, should be expected if a certain restriction is imposed upon the coefficients.
Let \(l\in \mathbb{N}\) be a fixed number and \(n=2l m-1+r,\) where \(m\in\mathbb{N},\) and \(r \in \{0,1,\ldots, 2l-1\}.\) The coefficients \(A=(a_j)_{j=0}^n\) are divided into \(2m\) blocks \(A_j=(a_{ l j}, a_{ l j+1}, \ldots, a_{ l j+l-1}), \ j=0, \ldots, 2m-1,\) of the length \(l,\) and the remaining coefficients \(\tilde{A}.\) Assume that \(\bigcup_{j=0}^{m-1} A_{2j} \cup \tilde{A}\) is a family of i.i.d. random variables with Gaussian distribution \( \mathcal{N}(0, \sigma^2),\) and \(a_{l(2j+1)+k}=a_{2lj+k}\) for any \(j=0, \ldots, m-1\) and \(k=0, \ldots, l-1,\) that is, \(A_{2j+1}=A_{2j}\). The author proves that under these assumptions \(\mathbb{E}[N_n(0, 2\pi)] = \frac{2n}{\sqrt{3}}+\mathcal{O}(n^{4/5}) \), as \(n \to \infty.\) In the case of only two equal blocks of coefficients, the author obtains the following asymptotic formula \(\mathbb{E}[N_n(0, 2\pi)] = \left(\frac{1}{2}+\frac{\sqrt{13}}{2\sqrt{3}}\right)n+\mathcal{O}(n^{4/5}), \) as \(n \to \infty,\) that is, in this case significantly more real zeros should be expected compared with those of the classical case.
MSC:
30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
30C99 Geometric function theory
12D10 Polynomials in real and complex fields: location of zeros (algebraic theorems)
26C10 Real polynomials: location of zeros
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