Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

On nonnegative cosine polynomials with nonnegative integral coefficients


Author: Mihail N. Kolountzakis
Journal: Proc. Amer. Math. Soc. 120 (1994), 157-163
MSC: Primary 42A05; Secondary 42A32
MathSciNet review: 1169037
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We show that there exist $ {p_0} > 0$ and $ {p_1}, \ldots ,{p_N}$ nonnegative integers, such that

$\displaystyle 0 \leqslant p(x) = {p_0} + {p_1}\cos x + \cdots + {p_N}\cos Nx$

and $ {p_0} \ll {s^{1/3}}$ for $ s = \sum\nolimits_{j = 0}^N {{p_j}} $, improving on a result of Odlyzko who showed the existence of such a polynomial $ p$ that satisfies $ {p_0} \ll {(s\log s)^{1/3}}$. Our result implies an improvement of the best known estimate for a problem of Erdős and Szekeres. As our method is nonconstructive, we also give a method for constructing an infinite family of such polynomials, given one good "seed" polynomial.

References [Enhancements On Off] (What's this?)

  • [1] N. Alon and J. Spencer, The probabilistic method, Wiley Interscience Series in Discrete Mathematics and Optimization, 1992. MR 1140703 (93h:60002)
  • [2] J. Beck, Flat polynomials on the unit circle--Note on a problem of Littlewood, Bull. London Math. Soc. 23 (1991), 269-277. MR 1123337 (93b:42002)
  • [3] J. Bourgain, Sur le minimum d'une somme de cosinus, Acta Arith. 45 (1986), 381-389. MR 847298 (87g:11096)
  • [4] P. Erdős and G. Szekeres, On the product $ \prod\nolimits_{k = 1}^n {(1 - {z^{{a_k}}})} $, Acad. Serbe Sci. Publ. Inst. Math. 13 (1959), 29-34. MR 0126425 (23:A3721)
  • [5] J.-P. Kahane, Some random series of functions, 2nd ed., Cambridge Stud. Adv. Math., vol. 5, Cambridge Univ. Press, Cambridge and New York, 1985. MR 833073 (87m:60119)
  • [6] L. Lovász, J. Spencer, and K. Vesztergombi, Discrepancy of set systems and matrices, European J. Combin. 7 (1986), 151-160. MR 856328 (88b:05036)
  • [7] A. M. Odlyzko, Minima of cosine sums and maxima of polynomials on the unit circle, J. London Math. Soc. (2) 26 (1982), 412-420. MR 684554 (84i:42001)
  • [8] K. F. Roth, On cosine polynomials corresponding to sets of integers, Acta Arith. 24 (1973), 347-355. MR 0342475 (49:7221)
  • [9] R. Salem and A. Zygmund, Some properties of trigonometric series whose terms have random signs, Acta Math. 91 (1954), 245-301 MR 0065679 (16:467b)
  • [10] J. Spencer, Six standard deviations suffice, Trans. Amer. Math. Soc. 289 (1985), 679-706. MR 784009 (86k:05004)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 42A05, 42A32

Retrieve articles in all journals with MSC: 42A05, 42A32


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1994-1169037-9
Article copyright: © Copyright 1994 American Mathematical Society