For a certain nonconvex optimisation problem, I need 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 too hard to see.

This problem is well studied for i.i.d. standard normal coefficients \(A(k),B(k)\). I need more general results; in particular I need to relax the identical distribution assumption.

It turns out there is some kind of point process result here, maybe impinging upon determinantal point processes.

Reading: [@FlascheExpected2017;@GarciaBrief2002;@VanderbeiComplex2015;@DumitrescuPositive2017;@EdelmanHow1995].

## Refs

- AnDP17: Angst, J., Dalmao, F., & Poly, G. (2017) On the real zeros of random trigonometric polynomials with dependent coefficients.
*ArXiv:1706.01654 [Math]*. - AzPh13: Azäis, J.-M., & Pham, V.-H. (2013) The record method for two and three dimensional parameters random fields.
*ArXiv:1302.1017 [Math]*. - AzWs09: Azäis, J.-M., & Wschebor, M. (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. DOI - Boyd07: Boyd, J. 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), 336–349. DOI - Das68: Das, M. (1968) The average number of real zeros of a random trigonometric polynomial.
*Mathematical Proceedings of the Cambridge Philosophical Society*, 64(3), 721–730. DOI - Dumi17: Dumitrescu, B. (2017)
*Positive trigonometric polynomials and signal processing applications*(Second edition.). Cham: Springer DOI - Dunn66: 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. DOI - EdKo95: Edelman, A., & Kostlan, E. (1995) How many zeros of a random polynomial are real?
*Bulletin of the American Mathematical Society*, 32(1), 1–38. DOI - Fara92: Farahmand, K. (1992) Number of real roots of a random trigonometric polynomial.
*Journal of Applied Mathematics and Stochastic Analysis*, 5(4), 307–313. DOI - FaLi10: Farahmand, K., & Li, T. (2010) Random Trigonometric Polynomials with Nonidentically Distributed Coefficients.
*International Journal of Stochastic Analysis*, 2010, 1–10. DOI - FaSa97: Farahmand, K., & Sambandham, M. (1997) On the expected number of real zeros of random trigonometric polynomials.
*Analysis*, 17(4), 345–354. DOI - Fara90: Farahmand, Kambiz. (1990) On the Average Number of Level Crossings of a Random Trigonometric Polynomial.
*The Annals of Probability*, 18(3), 1403–1409. - Flas17: Flasche, H. (2017) Expected number of real roots of random trigonometric polynomials.
*Stochastic Processes and Their Applications*, 127(12), 3928–3942. DOI - Garc02: García, A. G. (2002) A Brief Walk Through Sampling Theory.In P. W. Hawkes (Ed.), Advances in Imaging and Electron Physics (Vol. 124, pp. 63–137). Elsevier DOI
- HKPV09: Hough, J. B., Krishnapur, M., Peres, Y., & Vir’ag, B. (2009)
*Zeros of Gaussian Analytic Functions and Determinantal Point Processes*. American Mathematical Soc. - Kris06: Krishnapur, M. (2006) Zeros of Random Analytic Functions.
*ArXiv:Math/0607504*. - Megr03: Megretski, A. (2003) Positivity of trigonometric polynomials. In 42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475) (Vol. 4, pp. 3814–3817 vol.4). DOI
- PeRi13: Pemantle, R., & Rivin, I. (2013) The Distribution of Zeros of the Derivative of a Random Polynomial. In I. S. Kotsireas & E. V. Zima (Eds.), Advances in Combinatorics (pp. 259–273). Springer Berlin Heidelberg
- Schw91: Schweikard, A. (1991) Trigonometric polynomials with simple roots.
*Information Processing Letters*, 39(5), 231–236. DOI - SuSh12: Su, Z., & Shao, Q. (2012) Asymptotics of the variance of the number of real roots of random trigonometric polynomials.
*Science China Mathematics*, 55(11), 2347–2366. DOI - Vand15: Vanderbei, R. J. (2015) The Complex Roots of Random Sums.
*ArXiv:1508.05162 [Math]*. - Wilk91: Wilkins, J. E. (1991) Mean number of real zeros of a random trigonometric polynomial.
*Proceedings of the American Mathematical Society*, 111(3), 851–863. DOI