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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Palindromic random trigonometric polynomials


Authors: J. Brian Conrey, David W. Farmer and Özlem Imamoglu
Journal: Proc. Amer. Math. Soc. 137 (2009), 1835-1839
MSC (2000): Primary 60G99; Secondary 42A05, 30C15
Published electronically: December 15, 2008
MathSciNet review: 2470844
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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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Additional Information

J. Brian Conrey
Affiliation: Department of Mathematics, American Institute of Mathematics, 360 Portage Avenue, Palo Alto, California 94306
Email: conrey@aimath.org

David W. Farmer
Affiliation: Department of Mathematics, American Institute of Mathematics, 360 Portage Avenue, Palo Alto, California 94306
Email: farmer@aimath.org

Özlem Imamoglu
Affiliation: Department of Mathematics, Eidgen Technische Hochschule, CH-8092 Zurich, Switzerland
Email: ozlem@math.ethz.ch

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09776-1
PII: S 0002-9939(08)09776-1
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
Article copyright: © Copyright 2008 American Mathematical Society