×

zbMATH — the first resource for mathematics

Palindromic random trigonometric polynomials. (English) Zbl 1161.60017
Summary: We show that if a real trigonometric polynomial has few real roots, then the trigonometric polynomial obtained by writing the coefficients in reverse order must have many real roots. This is used to show that a class of random trigonometric polynomials has, on average, many real roots. In the case that the coefficients of a real trigonometric polynomial are independently and identically distributed, but with no other assumptions on the distribution, the expected fraction of real zeros is at least one-half. This result is best possible.

MSC:
60G99 Stochastic processes
42A05 Trigonometric polynomials, inequalities, extremal problems
30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
PDF BibTeX Cite
Full Text: DOI arXiv
References:
[1] P. Borwein, T. Erdélyi, R. Ferguson, and R. Lockhart, On the zeros of cosine polynomials: solution to a problem of Littlewood, Ann. of Math. (2) 167 (2008), no. 3, 1109 – 1117. · Zbl 1186.11045
[2] J. E. A. Dunnage, The number of real zeros of a random trigonometric polynomial, Proc. London Math. Soc. (3) 16 (1966), 53 – 84. · Zbl 0141.15003
[3] Alan Edelman and Eric Kostlan, How many zeros of a random polynomial are real?, Bull. Amer. Math. Soc. (N.S.) 32 (1995), no. 1, 1 – 37. · Zbl 0820.34038
[4] I. A. Ibragimov and N. B. Maslova, The mean number of real zeros of random polynomials. I. Coefficients with zero mean, Teor. Verojatnost. i Primenen. 16 (1971), 229 – 248 (Russian, with English summary). · Zbl 0277.60051
[5] I. A. Ibragimov and N. B. Maslova, The mean number of real zeros of random polynomials. II. Coefficients with a nonzero mean, Teor. Verojatnost. i Primenen. 16 (1971), 495 – 503 (Russian, with English summary).
[6] Ildar Ibragimov and Ofer Zeitouni, On roots of random polynomials, Trans. Amer. Math. Soc. 349 (1997), no. 6, 2427 – 2441. · Zbl 0872.30002
[7] B. R. Jamrom, The average number of real zeros of random polynomials, Dokl. Akad. Nauk SSSR 206 (1972), 1059 – 1060 (Russian). · Zbl 0269.60047
[8] M. Sambandham and N. Renganathan, On the number of real zeros of a random trigonometric polynomial: coefficients with nonzero mean, J. Indian Math. Soc. (N.S.) 45 (1981), no. 1-4, 193 – 203 (1984). · Zbl 0632.60063
[9] M. Sambandham and V. Thangaraj, On the average number of real zeros of a random trigonometric polynomial, J. Indian Math. Soc. (N.S.) 47 (1983), no. 1-4, 139 – 150 (1986). · Zbl 0605.60063
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.