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

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

Published electronically:
December 15, 2008

MathSciNet review:
2470844

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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

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