Zeros of random trigonometric polynomials

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For a certain nonconvex optimisation problem, I would like to know the expected number of real zeros of trigonometric polynomials

$0=\sum_{k=1}^{k=N}A(k)\sin(kx)B(k)\cos(kx)$

for given distributions over $$A(k)$$ and $$B(k)$$.

This is not exactly the usual sense of polynomial, although if one thinks about polynomials over the complex numbers and squint at it the relationship is not hard to see.

This problem is well studied for i.i.d. standard normal coefficients $$A(k),B(k)$$.

It turns out there are some determinantal point processes models for the distributions of zeros, which I should look into. (Ben Hough et al. 2009; Pemantle and Rivin 2013; Krishnapur 2006)

I need more general results than i.i.d. coefficients; in particular I need to relax the identical distribution assumption. đźš§

Refs

Angst, JĂĽrgen, Federico Dalmao, and Guillaume Poly. 2017. â€śOn the Real Zeros of Random Trigonometric Polynomials with Dependent Coefficients,â€ť June. http://arxiv.org/abs/1706.01654.

AzĂ¤is, Jean-Marc, and Viet-Hung Pham. 2013. â€śThe Record Method for Two and Three Dimensional Parameters Random Fields,â€ť February. http://arxiv.org/abs/1302.1017.

AzĂ¤is, Jean-Marc, and Mario Wschebor. 2009. Level Sets and Extrema of Random Processes and Fields: AzaĂŻs/Level Sets and Extrema of Random Processes and Fields. Hoboken, NJ, USA: John Wiley & Sons, Inc. https://doi.org/10.1002/9780470434642.

Ben Hough, John, Manjunath Krishnapur, Yuval Peres, and BĂˇlint VirĂˇg. 2009. Zeros of Gaussian Analytic Functions and Determinantal Point Processes. University Lecture Series, v. 51. Providence, R.I: American Mathematical Soc. http://math.iisc.ernet.in/~manju/GAF_book.pdf.

Boyd, John P. 2007. â€śComputing the Zeros of a Fourier Series or a Chebyshev Series or General Orthogonal Polynomial Series with Parity Symmetries.â€ť Computers & Mathematics with Applications 54 (3): 336â€“49. https://doi.org/10.1016/j.camwa.2007.01.015.

Das, Minaketan. 1968. â€śThe Average Number of Real Zeros of a Random Trigonometric Polynomial.â€ť Mathematical Proceedings of the Cambridge Philosophical Society 64 (3): 721â€“30. https://doi.org/10.1017/S0305004100043425.

Dumitrescu, Bogdan. 2017. Positive Trigonometric Polynomials and Signal Processing Applications. Second edition. Signals and Communication Technology. Cham: Springer. https://doi.org/10.1007/978-3-319-53688-0.

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Krishnapur, Manjunath. 2006. â€śZeros of Random Analytic Functions,â€ť July. http://arxiv.org/abs/math/0607504.

Krishnapur, Manjunath, and BĂˇlint VirĂˇg. 2014. â€śThe Ginibre Ensemble and Gaussian Analytic Functions.â€ť International Mathematics Research Notices 2014 (6): 1441â€“64. https://doi.org/10.1093/imrn/rns255.

Megretski, A. 2003. â€śPositivity of Trigonometric Polynomials.â€ť In 42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475), 4:3814â€“7 vol.4. https://doi.org/10.1109/CDC.2003.1271743.

Pemantle, Robin, and Igor Rivin. 2013. â€śThe Distribution of Zeros of the Derivative of a Random Polynomial.â€ť In Advances in Combinatorics, edited by Ilias S. Kotsireas and Eugene V. Zima, 259â€“73. Springer Berlin Heidelberg.

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Wilkins, J. Ernest. 1991. â€śMean Number of Real Zeros of a Random Trigonometric Polynomial.â€ť Proceedings of the American Mathematical Society 111 (3): 851â€“63. https://doi.org/10.1090/S0002-9939-1991-1039266-0.