Palindromic random trigonometric polynomials

Conrey, J.; Farmer, David; Imamoglu, Özlem

doi:10.1090/S0002-9939-08-09776-1

Palindromic random trigonometric polynomials
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by J. Brian Conrey, David W. Farmer and Özlem Imamoglu

Proc. Amer. Math. Soc. 137 (2009), 1835-1839

DOI: https://doi.org/10.1090/S0002-9939-08-09776-1

Published electronically: December 15, 2008

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Abstract:

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.

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