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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. 🚧

Refs

Angst, Jürgen, Federico Dalmao, and Guillaume Poly. 2017. “On the Real Zeros of Random Trigonometric Polynomials with Dependent Coefficients,” June. http://arxiv.org/abs/1706.01654.

Azäis, Jean-Marc, and Viet-Hung Pham. 2013. “The Record Method for Two and Three Dimensional Parameters Random Fields,” February. http://arxiv.org/abs/1302.1017.

Azäis, Jean-Marc, and Mario Wschebor. 2009. Level Sets and Extrema of Random Processes and Fields: Azaïs/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 Bálint Virág. 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. “Computing the Zeros of a Fourier Series or a Chebyshev Series or General Orthogonal Polynomial Series with Parity Symmetries.” Computers & Mathematics with Applications 54 (3): 336–49. https://doi.org/10.1016/j.camwa.2007.01.015.

Das, Minaketan. 1968. “The Average Number of Real Zeros of a Random Trigonometric Polynomial.” Mathematical Proceedings of the Cambridge Philosophical Society 64 (3): 721–30. 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.

Dunnage, J. E. A. 1966. “The Number of Real Zeros of a Random Trigonometric Polynomial.” Proceedings of the London Mathematical Society s3-16 (1): 53–84. https://doi.org/10.1112/plms/s3-16.1.53.

Edelman, Alan, and Eric Kostlan. 1995. “How Many Zeros of a Random Polynomial Are Real?” Bulletin of the American Mathematical Society 32 (1): 1–38. https://doi.org/10.1090/S0273-0979-1995-00571-9.

Farahmand, K. 1992. “Number of Real Roots of a Random Trigonometric Polynomial.” Journal of Applied Mathematics and Stochastic Analysis 5 (4): 307–13. https://doi.org/10.1155/S104895339200025X.

Farahmand, Kambiz. 1990. “On the Average Number of Level Crossings of a Random Trigonometric Polynomial.” The Annals of Probability 18 (3): 1403–9. https://www.jstor.org/stable/2244431.

Farahmand, K., and T. Li. 2010. “Random Trigonometric Polynomials with Nonidentically Distributed Coefficients.” International Journal of Stochastic Analysis 2010 (March): 1–10. https://doi.org/10.1155/2010/931565.

Farahmand, K., and M. Sambandham. 1997. “On the Expected Number of Real Zeros of Random Trigonometric Polynomials.” Analysis 17 (4): 345–54. https://doi.org/10.1524/anly.1997.17.4.345.

Flasche, Hendrik. 2017. “Expected Number of Real Roots of Random Trigonometric Polynomials.” Stochastic Processes and Their Applications 127 (12): 3928–42. https://doi.org/10.1016/j.spa.2017.03.018.

García, Antonio G. 2002. “A Brief Walk Through Sampling Theory.” In Advances in Imaging and Electron Physics, edited by Peter W. Hawkes, 124:63–137. Elsevier. https://doi.org/10.1016/S1076-5670(02)80042-8.

Krishnapur, Manjunath. 2006. “Zeros of Random Analytic Functions,” July. http://arxiv.org/abs/math/0607504.

Krishnapur, Manjunath, and Bálint Virág. 2014. “The Ginibre Ensemble and Gaussian Analytic Functions.” International Mathematics Research Notices 2014 (6): 1441–64. https://doi.org/10.1093/imrn/rns255.

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

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

Schweikard, Achim. 1991. “Trigonometric Polynomials with Simple Roots.” Information Processing Letters 39 (5): 231–36. https://doi.org/10.1016/0020-0190(91)90020-I.

Su, ZhongGen, and QiMan Shao. 2012. “Asymptotics of the Variance of the Number of Real Roots of Random Trigonometric Polynomials.” Science China Mathematics 55 (11): 2347–66. https://doi.org/10.1007/s11425-012-4525-5.

Vanderbei, Robert J. 2015. “The Complex Roots of Random Sums,” August. http://arxiv.org/abs/1508.05162.

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