Trigonometric polynomials with many real zeros and a Littlewood-type problem
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- by Peter Borwein and Tamás Erdélyi PDF
- Proc. Amer. Math. Soc. 129 (2001), 725-730
Abstract:
We examine the size of a real trigonometric polynomial of degree at most $n$ having at least $k$ zeros in $K := {\mathbb {R}} \ (\text {mod}\ 2\pi )$ (counting multiplicities). This result is then used to give a new proof of a theorem of Littlewood concerning flatness of unimodular trigonometric polynomials. Our proof is shorter and simpler than Littlewood’s. Moreover our constant is explicit in contrast to Littlewood’s approach, which is indirect.References
- József Beck, Flat polynomials on the unit circle—note on a problem of Littlewood, Bull. London Math. Soc. 23 (1991), no. 3, 269–277. MR 1123337, DOI 10.1112/blms/23.3.269
- P. Borwein, Some Old Problems on Polynomials with Integer Coefficients, in Approximation Theory IX, ed. C. Chui and L. Schumaker, Vanderbilt University Press (1998), 31–50.
- Peter Borwein and Tamás Erdélyi, Polynomials and polynomial inequalities, Graduate Texts in Mathematics, vol. 161, Springer-Verlag, New York, 1995. MR 1367960, DOI 10.1007/978-1-4612-0793-1
- P. Erdős, An inequality for the maximum of trigonometric polynomials, Ann. Polon. Math. 12 (1962), 151–154. MR 141933, DOI 10.4064/ap-12-2-151-154
- Jean-Pierre Kahane, Sur les polynômes à coefficients unimodulaires, Bull. London Math. Soc. 12 (1980), no. 5, 321–342 (French). MR 587702, DOI 10.1112/blms/12.5.321
- J. E. Littlewood, On the mean values of certain trigonometric polynomials, J. London Math. Soc. 36 (1961), 307–334. MR 141934, DOI 10.1112/jlms/s1-36.1.307
- J.E. Littlewood, On the mean value of certain trigonometric polynomials (II), Jour. London Math. Soc. 39 (1964), 511–552.
- J. E. Littlewood, The real zeros and value distributions of real trigonometrical polynomials, J. London Math. Soc. 41 (1966), 336–342. MR 196372, DOI 10.1112/jlms/s1-41.1.336
- J. E. Littlewood, On polynomials $\sum ^{n}\pm z^{m}$, $\sum ^{n}e^{\alpha _{m}i}z^{m}$, $z=e^{\theta _{i}}$, J. London Math. Soc. 41 (1966), 367–376. MR 196043, DOI 10.1112/jlms/s1-41.1.367
- John E. Littlewood, Some problems in real and complex analysis, D. C. Heath and Company Raytheon Education Company, Lexington, Mass., 1968. MR 0244463
- B. Saffari, Barker sequences and Littlewood’s “two-sided conjectures” on polynomials with $\pm 1$ coefficients, Séminaire d’Analyse Harmonique. Année 1989/90, Univ. Paris XI, Orsay, 1990, pp. 139–151. MR 1104693
Additional Information
- Peter Borwein
- Affiliation: Department of Mathematics and Statistics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6
- Email: pborwein@cecm.sfu.ca
- Tamás Erdélyi
- Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843
- Email: terdelyi@math.tamu.edu
- Received by editor(s): March 2, 1999
- Published electronically: November 3, 2000
- Additional Notes: The research of the first author was supported, in part, by NSERC of Canada. The research of the second author was supported, in part, by the NSF under Grant No. DMS–9623156.
- Communicated by: David R. Larson
- © Copyright 2000 Copyright retained by the authors
- Journal: Proc. Amer. Math. Soc. 129 (2001), 725-730
- MSC (2000): Primary 41A17
- DOI: https://doi.org/10.1090/S0002-9939-00-06021-4
- MathSciNet review: 1801998