Palindromic random trigonometric polynomials

Details Palindromic random trigonometric polynomials Palindromic random trigonometric polynomials

2008-12-09

arxiv.org/abs/0812.1752v1

Mathematics

Probability

5 pages. To appear in PAMS

Scientific paper

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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