Palindromic random trigonometric polynomials

2008/12/09 by Conrey, J. Brian, Farmer, David W., Imamoglu, Özlem
#30C15 #42A05 #60G99 #Complex Variables (math.CV) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.0812.1752

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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