Palindromic random trigonometric polynomials
Authors:
J. Brian Conrey, David W. Farmer and Özlem Imamoglu
Journal:
Proc. Amer. Math. Soc. 137 (2009), 18351839
MSC (2000):
Primary 60G99; Secondary 42A05, 30C15
Published electronically:
December 15, 2008
MathSciNet review:
2470844
Fulltext 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 onehalf. This result is best possible.
 1.
P.
Borwein, T.
Erdélyi, R.
Ferguson, and R.
Lockhart, On the zeros of cosine polynomials: solution to a problem
of Littlewood, Ann. of Math. (2) 167 (2008),
no. 3, 1109–1117. MR 2415396
(2009d:42001), http://dx.doi.org/10.4007/annals.2008.167.1109
 2.
J.
E. A. Dunnage, The number of real zeros of a random trigonometric
polynomial, Proc. London Math. Soc. (3) 16 (1966),
53–84. MR
0192532 (33 #757)
 3.
Alan
Edelman and Eric
Kostlan, How many zeros of a random polynomial
are real?, Bull. Amer. Math. Soc. (N.S.)
32 (1995), no. 1,
1–37. MR
1290398 (95m:60082), http://dx.doi.org/10.1090/S027309791995005719
 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 (Russian, with English summary).
MR
0286157 (44 #3371)
 5.
I.
A. Ibragimov and N.
B. Maslova, The mean number of real zeros of random polynomials.
II. Coefficients with a nonzero mean, Teor. Verojatnost. i Primenen.
16 (1971), 495–503 (Russian, with English summary).
MR
0288824 (44 #6019)
 6.
Ildar
Ibragimov and Ofer
Zeitouni, On roots of random
polynomials, Trans. Amer. Math. Soc.
349 (1997), no. 6,
2427–2441. MR 1390040
(97h:60050), http://dx.doi.org/10.1090/S0002994797017662
 7.
B.
R. Jamrom, The average number of real zeros of random
polynomials, Dokl. Akad. Nauk SSSR 206 (1972),
1059–1060 (Russian). 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 (1981), no. 14, 193–203 (1984). 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 (1983), no. 14, 139–150 (1986). MR 878090
(87m:42001)
 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, 11091117. MR 2415396
 2.
 J.E.A. Dunnage, The number of real zeros of a random trigonometric polynomial, Proc. London. Math. Soc. 16 (1966), 5384. 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), 137. 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), 229248. 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), 495503. MR 0288824 (44:6019)
 6.
 I.A. Ibragimov and O. Zeitouni, On roots of random polynomials, Trans. Amer. Math. Soc. 349 (1997), no. 6, 24272441. MR 1390040 (97h:60050)
 7.
 B.R. Jamrom, The average number of real zeros of random polynomials. Soviet Math. Dokl. 13 (1972), no. 5, 13811383. 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), 193203. 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. 14, 139150. 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, CH8092 Zurich, Switzerland
Email:
ozlem@math.ethz.ch
DOI:
http://dx.doi.org/10.1090/S0002993908097761
PII:
S 00029939(08)097761
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
