|
Palindromic random trigonometric polynomials
Author(s):
J.
Brian
Conrey;
David
W.
Farmer;
Özlem
Imamoglu
Journal:
Proc. Amer. Math. Soc.
137
(2009),
1835-1839.
MSC (2000):
Primary 60G99;
Secondary 42A05, 30C15
Posted:
December 15, 2008
Retrieve article in:
PDF
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.
References:
-
- 1.
- P. Borwein, T. Erdélyi, R. Ferguson, and R. Lockhart, On the zeros of cosine polynomials: solution to a problem of Littlewood. Annals of Math. (2) 167 (2008), no. 3, 1109-1117. MR 2415396
- 2.
- J.E.A. Dunnage, The number of real zeros of a random trigonometric polynomial, Proc. London. Math. Soc. 16 (1966), 53-84. MR 0192532 (33:757)
- 3.
- A. Edelman and E. Kostlan, How many zeros of a random polynomial are real? Bull. Amer. Math. Soc. 32 (1995), 1-37. MR 1290398 (95m:60082)
- 4.
- I.A. Ibragimov and N.B. Maslova, The mean number of real zeros of random polynomials. I: Coefficients with zero mean. Teor. Verojatnost. i Primenen. 16 (1971), 229-248. MR 0286157 (44:3371)
- 5.
- I.A. Ibragimov and N.B. Maslova, The mean number of real zeros of random polynomials. II: Coefficients with nonzero mean. Teor. Verojatnost. i Primenen. 16 (1971), 495-503. MR 0288824 (44:6019)
- 6.
- I.A. Ibragimov and O. Zeitouni, On roots of random polynomials, Trans. Amer. Math. Soc. 349 (1997), no. 6, 2427-2441. MR 1390040 (97h:60050)
- 7.
- B.R. Jamrom, The average number of real zeros of random polynomials. Soviet Math. Dokl. 13 (1972), no. 5, 1381-1383. MR 0314114 (47:2666)
- 8.
- M. Sambandham and N. Renganathan, On the number of real zeros of a random trigonometric polynomial: Coefficients with nonzero mean. J. Indian Math. Soc. (N.S.) 45 (1984), 193-203. MR 828871 (87g:42003)
- 9.
- M. Sambandham and V. Thangaraj, On the average number of real zeros of a random trigonometric polynomial. J. Indian Math. Soc. (N.S.) 47 (1986), no. 1-4, 139-150. MR 878090 (87m:42001)
Similar Articles:
Retrieve articles in Proceedings of the American Mathematical Society
with MSC
(2000):
60G99,
42A05, 30C15
Retrieve articles in all Journals with MSC
(2000):
60G99,
42A05, 30C15
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:
10.1090/S0002-9939-08-09776-1
PII:
S 0002-9939(08)09776-1
Received by editor(s):
August 12, 2008
Posted:
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 of article:
Copyright
2008,
American Mathematical Society
|
Comments: webmaster@ams.org