## 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.## References

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## Bibliographic Information

**J. Brian Conrey**- Affiliation: Department of Mathematics, American Institute of Mathematics, 360 Portage Avenue, Palo Alto, California 94306
- MR Author ID: 51070
- Email: conrey@aimath.org
**David W. Farmer**- Affiliation: Department of Mathematics, American Institute of Mathematics, 360 Portage Avenue, Palo Alto, California 94306
- MR Author ID: 341467
- Email: farmer@aimath.org
**Özlem Imamoglu**- Affiliation: Department of Mathematics, Eidgen Technische Hochschule, CH-8092 Zurich, Switzerland
- Email: ozlem@math.ethz.ch
- Received by editor(s): August 12, 2008
- Published electronically: December 15, 2008
- Additional Notes: The research of the first two authors was supported by the American Institute of Mathematics and the National Science Foundation
- Communicated by: Richard C. Bradley
- © Copyright 2008 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**137**(2009), 1835-1839 - MSC (2000): Primary 60G99; Secondary 42A05, 30C15
- DOI: https://doi.org/10.1090/S0002-9939-08-09776-1
- MathSciNet review: 2470844