Abstract

We consider the Banach space of two homogeneous polynomials endowed with the supremum norm ∥⋅∥D(β)\|\cdot\|_{D(\beta)} over circle sectors D(β)D(\beta) of angle β\beta for several values of β∈[0,2π]\beta\in[0,2\pi]. We provide an explicit formula for ∥⋅∥D(β)\|\cdot\|_{D(\beta)}, a full description of the extreme points of the corresponding unit balls, and a parametrization and a plot of their unit spheres. This work is an extension of a series of papers on the same topic published in the last decade and it has a number of applications to obtain polynomial-type inequalities.

Author information

Contact details are reproduced from the original publication and may be historical.

Gustavo A. Muñoz-Fernández

Dep. de Análisis Matemático, Universidad Complutense de Madrid, Plaza Ciencias 3, 28040 Madrid, Spain

gustavo_fernandez@mat.ucm.es

Daniel Pellegrino

Dep. de Matemática, Universidade Federal da Paraíba, 58.051-900 - João Pessoa, Brazil

pellegrino@pq.cnpq.br

Juan B. Seoane-Sepúlveda

Dep. de Análisis Matemático, Universidad Complutense de Madrid, Plaza Ciencias 3, 28040 Madrid, Spain

jseoane@mat.ucm.es

Andreas Weber

Institut für Algebra und Geometrie, Karlsruher Institut für Technologie, Kaiserstr. 89-93, 76128 Karlsruhe, Germany

andreasweber.mail@gmail.com

Keywords

  • Bernstein and Markov inequalities
  • unconditional constants
  • polarizations constants
  • polynomial inequalities
  • homogeneous polynomials
  • extreme points

Mathematics Subject Classification

46G25; 46B28, 41A44

References

  1. R. M. Aron, M. Klimek: Supremum norms for quadratic polynomials, Arch. Math. 76 (2001) 73–80.https://doi.org/10.1007/s000130050544Google Scholar
  2. L. Białas-Cież, P. Goetgheluck: Constants in Markov’s inequality on convex sets, East J. Approx. 1(3) (1995) 379–389.Google Scholar
  3. Y. S. Choi, S. G. Kim: The unit ball of P(2 l22), Arch. Math. 71(6) (1998) 472–480.https://doi.org/10.1007/s000130050292Google Scholar
  4. Y. S. Choi, S. G. Kim: Smooth points of the unit ball of the space P(2 l1), Results Math. 36 (1999) 26–33.Google Scholar
  5. Y. S. Choi, S. G. Kim: Exposed points of the unit balls of the spaces P(2 lp2) (p = 1, 2, ∞), Indian J. Pure Appl. Math. 35 (2004) 37–41.Google Scholar
  6. J. L. Gámez-Merino, G. A. Muñoz-Fernández, V. M. Sánchez, J. B. Seoane-Sepúlveda: Inequalities for polynomials on the unit square via the Krein-Milman Theorem, J. Convex Analysis 20(1) (2013) 125–142.https://doi.org/10.68381/jca20008Google Scholar
  7. B. C. Grecu: Geometry of homogeneous polynomials on two-dimensional real Hilbert spaces, J. Math. Anal. Appl. 293(1) (2004) 578–588.https://doi.org/10.1016/j.jmaa.2004.01.020Google Scholar
  8. B. C. Grecu: Extreme 2-homogeneous polynomials on Hilbert spaces, Quaest. Math. 25(4)(2002) 421–435.https://doi.org/10.2989/16073600209486027Google Scholar
  9. B. C. Grecu: Geometry of 2-homogeneous polynomials on lp spaces, 1 < p < ∞, J. Math. Anal. Appl. 273(1) (2002) 262–282.https://doi.org/10.1016/S0022-247X(02)00217-2Google Scholar
  10. B. C. Grecu: Smooth 2-homogeneous polynomials on Hilbert spaces, Arch. Math. 76(6) (2001) 445–454.https://doi.org/10.1007/PL00000456Google Scholar
  11. B. C. Grecu: Geometry of three-homogeneous polynomials on real Hilbert spaces, J. Math. Anal. Appl. 246(1) (2000) 217–229.https://doi.org/10.1006/jmaa.2000.6783Google Scholar
  12. B. C. Grecu, G. A. Muñoz-Fernández, J. B. Seoane-Sepúlveda: The unit ball of the complex P(3 H), Math. Z. 263 (2009) 775–785.https://doi.org/10.1007/s00209-008-0438-yGoogle Scholar
  13. B. C. Grecu, G. A. Muñoz-Fernández, J. B. Seoane-Sepúlveda: Unconditional constants and polynomial inequalities, J. Approx. Theory 161(2) (2009) 706–722.https://doi.org/10.1016/j.jat.2008.12.001Google Scholar
  14. A. G. Konheim, T. J. Rivlin: Extreme points of the unit ball in a space of real polynomials, Amer. Math. Monthly 73 (1966) 505–507.https://doi.org/10.2307/2315472Google Scholar
  15. L. Milev, S. G. Révész: Bernstein’s inequality for multivariate polynomials on the standard simplex, J. Inequal. Appl. 2005(2) (2005) 145–163.https://doi.org/10.1155/JIA.2005.145Google Scholar
  16. L. Milev, N. Naidenov: Strictly definite extreme points of the unit ball in a polynomial space, C. R. Acad. Bulg. Sci. 61 (2008) 1393–1400.Google Scholar
  17. L. Milev, N. Naidenov: Indefinite extreme points of the unit ball in a polynomial space, Acta Sci. Math. 77 (2011) 409–424.https://doi.org/10.1007/BF03643924Google Scholar
  18. G. A. Muñoz-Fernández, D. Pellegrino, J. Ramos Campos, J. B. Seoane-Sepúlveda: On the optimality of the hypercontractivity of the complex Bohnenblust-Hille inequality, arXiv:1301.1539 (2013).Google Scholar
  19. G. A. Muñoz-Fernández, S. G. Révész, J. B. Seoane-Sepúlveda: Geometry of homogeneous polynomials on non symmetric convex bodies, Math. Scand. 105 (2009) 147–160.https://doi.org/10.7146/math.scand.a-15111Google Scholar
  20. G. A. Muñoz-Fernández, V. M. Sánchez, J. B. Seoane-Sepúlveda: Estimates on the derivative of a polynomial with a curved majorant using convex techniques, J. Convex Analysis 17(1) (2010) 241–252.https://doi.org/10.68381/jca17018Google Scholar
  21. G. A. Muñoz-Fernández, V. M. Sánchez, J. B. Seoane-Sepúlveda: Lp -analogues of Bernstein and Markov inequalities, Math. Inequal. Appl. 14(1) (2011) 135–145.https://doi.org/10.7153/mia-14-11Google Scholar
  22. G. A. Muñoz-Fernández, Y. Sarantopoulos: Bernstein and Markov-type inequalities for polynomials in real Banach spaces, Math. Proc. Camb. Philos. Soc. 133 (2002) 515–530.https://doi.org/10.1017/S0305004102006217Google Scholar
  23. G. A. Muñoz-Fernández, Y. Sarantopoulos, J. B. Seoane-Sepúlveda: An application of the Krein-Milman theorem to Bernstein and Markov inequalities, J. Convex Analysis 15 (2008) 299–312.https://doi.org/10.68381/jca15021Google Scholar
  24. G. A. Muñoz-Fernández, J. B. Seoane-Sepúlveda: Geometry of Banach spaces of Trinomials, J. Math. Anal. Appl. 340 (2008) 1069–1087.https://doi.org/10.1016/j.jmaa.2007.09.010Google Scholar
  25. D. Nadzhmiddinov, Yu. N. Subbotin: Markov inequalities for polynomials on triangles, Mat. Zametki 46(2) (1989) 76–82, 159 (in Russian); Math. Notes 46(2) (1989) 627–631 (in English).https://doi.org/10.1007/BF01137627Google Scholar
  26. S. Neuwirth: The maximum modulus of a trigonometric trinomial, J. Anal. Math. 104 (2008) 371–396.https://doi.org/10.1007/s11854-008-0028-2Google Scholar
  27. D. R. Wilhelmsen: A Markov inequality in several dimensions, J. Approx. Theory 11 (1974) 216–220.https://doi.org/10.1016/0021-9045(74)90012-4Google Scholar

Suggested citation

G. A. Muñoz-Fernández, D. Pellegrino, J. B. Seoane-Sepúlveda, A. Weber. “Supremum Norms for 2-Homogeneous Polynomials on Circle Sectors.” Journal of Convex Analysis 21 (2014), No. 3, 745–764. https://doi.org/10.68381/jca21040

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2014. Rights now held by Banach Press.