Large Sieve Inequalities via Subharmonic Methods and the Mahler Measure of the Fekete Polynomials

Erdélyi, T. ; Lubinsky, D. S.

Canadian journal of mathematics, Tome 59 (2007) no. 4, pp. 730-741

Résumé

We investigate large sieve inequalities such as $$\frac{1}{m}\underset{j=1}{\overset{m}{\mathop{\sum }}}\,\,\psi \left( \log \,|P({{e}^{i\tau j}})| \right)\,\le \,\frac{C}{2\pi }\,\int_{0}^{2\pi }{\psi }\,\left( \log [e\,|P({{e}^{i\tau }})|] \right)\,d\tau ,$$ where $\psi$ is convex and increasing, $P$ is a polynomial or an exponential of a potential, and the constant $C$ depends on the degree of $P$ , and the distribution of the points $0\,\le \,{{\tau }_{1}}\,<\,{{\tau }_{2}}\,<\,\cdots \,<\,{{\tau }_{m}}\,\le \,2\pi$ . The method allows greater generality and is in some ways simpler than earlier ones. We apply our results to estimate the Mahler measure of Fekete polynomials.

Détail

DOI : 10.4153/CJM-2007-032-x

Mots-clés : 41A17

Bibliographie

[1] [1] Beller, E. and Newman, D. J., An extremal problem for the geometric mean of polynomials. Proc. Amer. Math. Soc. 39(1973), 313–317. Google Scholar

[2] [2] Borwein, P., Computational Excursions in Analysis and Number Theory. CMS Books in Mathematics 10, Springer-Verlag, New York, 2002. Google Scholar

[3] [3] Borwein, P. and Erdélyi, T., Polynomials and Polynomial Inequalities. Graduate Texts in Mathematics 161, Springer-Verlag, New York, 1995. Google Scholar

[4] [4] Borwein, P. and Erdélyi, T., On the zeros of polynomials with restricted coefficients. Illinois J. Math. 41(1997), no. 4, 667–675. Google Scholar

[5] [5] Boyd, D., On a problem of Byrnes concerning polynomials with restricted coefficients. Math. Comput. 66(1997), no. 220, 1697–1703. Google Scholar

[6] [6] Davis, P. J. and Rabinowitz, P., Methods of Numerical Integration. 2nd edition. Academic Press, Orlando, FL, 1984. Google Scholar

[7] [7] Erdélyi, T., Li, X., and Saff, E. B., Remez- and Nikolskii-type inequalities for logarithmic potentials. SIAM J. Math. Anal. 25(1994), 365–383. Google Scholar

[8] [8] Golinskii, L., Lubinsky, D. S., and Nevai, P., Large sieve estimates on arcs of a circle. J. Number Theory 91(2001), no. 2, 206–229. Google Scholar

[9] [9] Kobindarajah, C. K. and Lubinsky, D. S., Marcinkiewicz-Zygmund inequalities for all arcs of the circle. In: Advances in Constructive Approximation, Nashboro Press, Brentwood, TN, 2004, pp. 255–264. Google Scholar

[10] [10] Littlewood, J. E., The real zeros and value distributions of real trigonometric polynomials. J. London Math. Soc. 41(1966), 336–342. Google Scholar

[11] [11] Lubinsky, D. S., Marcinkiewicz-Zygmund inequalities: methods and results. In: Recent Progress in Inequalities, Math. Appl. 430, Kluwer Academic Publishers, Dordrecht, 1998, pp. 213–240. Google Scholar

[12] [12] Lubinsky, D. S., Mate, A., and Nevai, P., Quadrature sums involving pth powers of polynomials. SIAM J. Math. Anal. 18(1987), no. 2, 531–544. Google Scholar

[13] [13] Mastroianni, G. and Russo, M. G., Weighted Marcinkiewicz inequalities and boundedness of the Lagrange operator. In: Mathematical Analysis and Applications, Hadronic Press, Palm Harbor, FL, 2000, pp. 149–182. Google Scholar

[14] [14] Montgomery, H. L., The analytic principle of the large sieve. Bull. Amer. Math. Soc. 84(1978), no. 4, 547–567. Google Scholar

[15] [15] Ransford, T., Potential Theory in the Complex Plane. London Mathematical Society Student Texts 28, Cambridge University Press, Cambridge, 1995. Google Scholar

[16] [16] Zhong, L. and Zhu, L. Y., The Marcinkiewicz-Zygmund inequality on a smooth simple arc. J. Approx. Theory 83(1995), no. 1, 65–83. Google Scholar