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    Erdélyi, Tamás 2015. Flatness of conjugate reciprocal unimodular polynomials. Journal of Mathematical Analysis and Applications, Vol. 432, Issue. 2, p. 699.

  • Mathematical Proceedings of the Cambridge Philosophical Society, Volume 151, Issue 2
  • September 2011, pp. 373-384

On the Lq norm of cyclotomic Littlewood polynomials on the unit circle

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  • Published online: 13 July 2011

Let n be the collection of all (Littlewood) polynomials of degree n with coefficients in {−1, 1}. In this paper we prove that if (P) is a sequence of cyclotomic polynomials P, then for every q > 2 with some a = a(q) > 1/2 depending only on q, where The case q = 4 of the above result is due to P. Borwein, Choi and Ferguson. We also prove that if (P) is a sequence of cyclotomic polynomials P, then for every 0 < q < 2 with some 0 < b = b(q) < 1/2 depending only on q. Similar results are conjectured for Littlewood polynomials of odd degree. Our main tool here is the Borwein–Choi Factorization Theorem.

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[Bo]P. Borwein Computational excursions in analysis and number theory (Springer-Verlag, 2002).

[BC]P. Borwein and K.S. Choi On cyclotomic polynomials with ± 1 coefficients. Experiment. Math. 8 (1995), 399407.

[BE]P. Borwein and T. Erdélyi Polynomials and polynomial inequalities (Springer-Verlag, 1995).

[Em]K. M. Eminyan The L1 norm of a trigonometric sum. Math. Notes 76 (2004), 124132.

[Er2]T. Erdélyi How far is a sequence of ultraflat unimodular polynomials from being conjugate reciprocal. Michigan Math. J. 49 (2001), 259264.

[Er3]T. Erdélyi A proof of Saffari's “near-orthogonality" conjecture for ultraflat sequences of unimodular polynomials. C. R. Acad. Sci. Paris Sér. I Math. 333 (2001), 623628.

[Ka]J.P. Kahane Sur les polynomes a coefficient unimodulaires. Bull. London Math. Soc. 12 (1980), 321342.

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[QS]H. Queffelec and B. Saffari On Bernstein's inequality and Kahane's ultraflat polynomials. J. Fourier Anal. Appl. 2 (1996, 6), 519582.

[Sa]B. Saffari The phase behavior of ultraflat unimodular polynomials. In Probabilistic and Stochastic Methods in Analysis, with Applications (Kluwer Academic Publishers, 1992), 555572.

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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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