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

Published online by Cambridge University Press:  13 July 2011

TAMÁS ERDÉLYI*
Affiliation:
Department of Mathematics, Texas A&M University, College Station, Texas 77843, U.S.A. e-mail: terdelyi@math.tamu.edu

Abstract

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.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 2011

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References

REFERENCES

[Be]Beck, J. “Flat” polynomials on the unit circle – note on a problem of Littlewood. Bull. London Math. Soc. (1991), 269–277.CrossRefGoogle Scholar
[BB]Bombieri, E. and Bourgain, J.On Kahane's ultraflat polynomials. J. Eur. Math. Soc. 11 (2009, 3), 627703.Google Scholar
[Bo]Borwein, P.Computational excursions in analysis and number theory (Springer-Verlag, 2002).CrossRefGoogle Scholar
[BC]Borwein, P. and Choi, K.S.On cyclotomic polynomials with ± 1 coefficients. Experiment. Math. 8 (1995), 399407.Google Scholar
[BCF]Borwein, P., Choi, K. S. and Ferguson, R.Norm of Littlewood cyclotomic polynomials. Math. Proc. Camb. Phil. Soc. 138 (2005), 315326.Google Scholar
[BE]Borwein, P. and Erdélyi, T.Polynomials and polynomial inequalities (Springer-Verlag, 1995).Google Scholar
[BLM]Brillhart, J., Lomont, J.S. and Morton, P.Cyclotomic properties of the Rudin–Shapiro polynomials. J. Reine Angew. Math. 288 (1976), 3775.Google Scholar
[Em]Eminyan, K. M.The L 1 norm of a trigonometric sum. Math. Notes 76 (2004), 124132.CrossRefGoogle Scholar
[Er1]Erdélyi, T.The phase problem of ultraflat unimodular polynomials: the resolution of the conjecture of Saffari. Math. Ann. 321 (2001), 905924.Google Scholar
[Er2]Erdélyi, T.How far is a sequence of ultraflat unimodular polynomials from being conjugate reciprocal. Michigan Math. J. 49 (2001), 259264.Google Scholar
[Er3]Erdélyi, T.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.Google Scholar
[Er4]Erdélyi, T.On the real part of ultraflat sequences of unimodular polynomials: consequences implied by the resolution of the Phase Problem. Math. Ann. 326 (2003), 489498.Google Scholar
[Er]Erdős, P.Some unsolved problems. Michigan Math. J. 4 (1957), 291300.CrossRefGoogle Scholar
[GR]Green, B. and Ruzsa, I. Z.On the Hardy–Littlewood majorant problem. Math. Proc. Camb. Phil. Soc. 137 (2004), 511517.Google Scholar
[Ka]Kahane, J.P.Sur les polynomes a coefficient unimodulaires. Bull. London Math. Soc. 12 (1980), 321342.Google Scholar
[Ko]Körner, T.On a polynomial of J.S. Byrnes. Bull. London Math. Soc. 12 (1980), 219224.CrossRefGoogle Scholar
[Li1]Littlewood, J. E.On polynomials ∑ ± z m, ∑exp(αmi)z m, z = e iθ. J. London Math. Soc. 41 (1966), 367376.Google Scholar
[Li2]Littlewood, J. E. Some problems in real and complex analysis. Heath Mathematical Monographs (Lexington, Massachusetts, 1968).Google Scholar
[QS]Queffelec, H. and Saffari, B.On Bernstein's inequality and Kahane's ultraflat polynomials. J. Fourier Anal. Appl. 2 (1996, 6), 519582.Google Scholar
[Sa]Saffari, B. The phase behavior of ultraflat unimodular polynomials. In Probabilistic and Stochastic Methods in Analysis, with Applications (Kluwer Academic Publishers, 1992), 555572.Google Scholar