DOI QRμ½”λ“œ

DOI QR Code

POLARIZATION AND UNCONDITIONAL CONSTANTS OF π“Ÿ(2d*(1,Ο‰)2)

  • Kim, Sung Guen (Department of Mathematics Kyungpook National University)
  • Received : 2014.02.02
  • Published : 2014.07.31

Abstract

We explicitly calculate the polarization and unconditional constants of $\mathcal{P}(^2d_*(1,{\omega})^2)$.

Keywords

extreme 2-homogeneous polynomials;the predual of two dimensional Lorentz sequence space;the polarization constant;the unconditional constant

References

  1. Y. S. Choi and S. G. Kim, Exposed points of the unit balls of the spaces P($^2l^2_p$) (p = 1, 2, ${\infty}$), Indian J. Pure Appl. Math. 35 (2004), no. 1, 37-41.
  2. R. M. Aron, Y. S. Choi, S. G. Kim and M. Maestre, Local properties of polynomials on a Banach space, Illinois J. Math. 45 (2001), no. 1, 25-39.
  3. Y. S. Choi, H. Ki, and S. G. Kim, Extreme polynomials and multilinear forms on $l_1$, J. Math. Anal. Appl. 228 (1998), no. 2, 467-482. https://doi.org/10.1006/jmaa.1998.6161
  4. Y. S. Choi and S. G. Kim, The unit ball of P($^2l^2_2$), Arch. Math. (Basel) 71 (1998), no. 6, 472-480. https://doi.org/10.1007/s000130050292
  5. Y. S. Choi and S. G. Kim, Extreme polynomials on $c_0$, Indian J. Pure Appl. Math. 29 (1998), no. 10, 983-989.
  6. Y. S. Choi and S. G. Kim, Smooth points of the unit ball of the space P($^2l_1$), Results Math. 36 (1999), no. 1-2, 26-33. https://doi.org/10.1007/BF03322099
  7. B. C. Grecu, Geometry of 2-homogeneous polynomials on $l_p$ spaces, 1 < p < ${\infty}$, J. Math. Anal. Appl. 273 (2002), no. 2, 262-282.
  8. S. Dineen, Complex Analysis on Infinite Dimensional Spaces, Springer-Verlag, London, 1999.
  9. S. Dineen, Extreme integral polynomials on a complex Banach space, Math. Scand. 92 (2003), no. 1, 129-140. https://doi.org/10.7146/math.scand.a-14397
  10. J. L. Gamez-Merino, G. A. Munoz-Fernandez, V. M. Sanchez, and J. B. Seoane-Sepulveda, Inequalities for polynomials on the unit square via the Krein-Milman Theorem, J. Convex Anal. 20 (2013), no. 1, 125-142.
  11. S. G. Kim, The unit ball of P($^2d{\ast}(1,w)^2$), Math. Proc. R. Ir. Acad. 111A (2011), no. 2, 79-94.
  12. B. C. Grecu, G. A. Munoz-Fernandez, and J. B. Seoane-Sepulveda, Unconditional constants and polynomial inequalities, J. Approx. Theory 161 (2009), no. 2, 706-722. https://doi.org/10.1016/j.jat.2008.12.001
  13. S. G. Kim, Exposed 2-homogeneous polynomials on P($^2l^2_p$) ($1\;{\leq}p\;{\leq}\;{\infty}$), Math. Proc. R. Ir. Acad. 107 (2007), no. 2, 123-129. https://doi.org/10.3318/PRIA.2007.107.2.123
  14. S. G. Kim, The unit ball of $L_s$($^2l^2_{\infty}$), Extracta Math. 24 (2009), no. 1, 17-29.
  15. S. G. Kim, Smooth polynomials of P($^2d{\ast}(1,w)^2$), Math. Proc. R. Ir. Acad. 113A (2013), no. 1, 45-58.
  16. S. G. Kim, The unit ball of $L_s(^2d{\ast}(1,w)^2)$, Kyungpook Math. J. 53 (2013), no. 2, 295-306. https://doi.org/10.5666/KMJ.2013.53.2.295
  17. S. G. Kim and S. H. Lee, Exposed 2-homogeneous polynomials on Hilbert spaces, Proc. Amer. Math. Soc. 131 (2003), no. 2, 449-453. https://doi.org/10.1090/S0002-9939-02-06544-9
  18. J. Lee and K. S. Rim, Properties of symmetric matrices, J. Math. Anal. Appl. 305 (2005), no. 1, 219-226. https://doi.org/10.1016/j.jmaa.2004.11.011
  19. L. Milev and N. Naidenov, Strictly definite extreme points of the unit ball in a polynomial space, C. R. Acad. Bulgare Sci. 61 (2008), no. 11, 1393-1400.
  20. G. A. Munoz-Fernandez, S. Revesz, and J. B. Seoane-Sepulveda, Geometry of homoge-neous polynomials on non symmetric convex bodies, Math. Scand. 105 (2009), no. 1, 147-160. https://doi.org/10.7146/math.scand.a-15111
  21. G. A. Munoz-Fernandez and J. B. Seoane-Sepulveda, Geometry of Banach spaces of trinomials, J. Math. Anal. Appl. 340 (2008), no. 2, 1069-1087. https://doi.org/10.1016/j.jmaa.2007.09.010
  22. S. Revesz and Y. Sarantopoulos, Plank problems, polarization and Chebyshev constants, J. Korean Math. Soc. 41 (2004), no. 1, 157-174. https://doi.org/10.4134/JKMS.2004.41.1.157
  23. R. A. Ryan and B. Turett, Geometry of spaces of polynomials, J. Math. Anal. Appl. 221 (1998), no. 2, 698-711. https://doi.org/10.1006/jmaa.1998.5942

Cited by

  1. Exposed 2-Homogeneous Polynomials on the two-Dimensional Real Predual of Lorentz Sequence Space vol.13, pp.5, 2016, https://doi.org/10.1007/s00009-015-0658-4
  2. Extreme bilinear forms on $$\mathbb {R}^n$$Rn with the supremum norm vol.77, pp.2, 2018, https://doi.org/10.1007/s10998-018-0246-z