Abstract and Applied Analysis

Extension of The Best Approximation Operator in Orlicz Spaces

Ivana Carrizo, Sergio Favier, and Felipe Zó

Full-text: Open access

Abstract

Let ( Ω , 𝒜 , μ ) be a probability space and 𝒜 a sub- σ -lattice of the σ -algebra 𝒜 . We study an extension of the best ϕ -approximation operator from an Orlicz space L ϕ to the space L ϕ , where ϕ denotes the derivative of the convex, but not necessarily a strictly convex function ϕ . We obtain convergence results when a sequence of σ -algebras n converges to in a suitable way.

Article information

Source
Abstr. Appl. Anal., Volume 2008 (2008), Article ID 374742, 15 pages.

Dates
First available in Project Euclid: 9 September 2008

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1220969155

Digital Object Identifier
doi:10.1155/2008/374742

Mathematical Reviews number (MathSciNet)
MR2393120

Zentralblatt MATH identifier
1160.41308

Citation

Carrizo, Ivana; Favier, Sergio; Zó, Felipe. Extension of The Best Approximation Operator in Orlicz Spaces. Abstr. Appl. Anal. 2008 (2008), Article ID 374742, 15 pages. doi:10.1155/2008/374742. https://projecteuclid.org/euclid.aaa/1220969155


Export citation

References

  • D. Landers and L. Rogge, “Best approximants in $L_\phi$-spaces,” Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, vol. 51, no. 2, pp. 215–237, 1980.
  • D. Landers and L. Rogge, “Isotonic approximation in $L_s$,” Journal of Approximation Theory, vol. 31, no. 3, pp. 199–223, 1981.
  • F. Mazzone and H. Cuenya, “Isotonic approximations in $L_1$,” Journal of Approximation Theory, vol. 117, no. 2, pp. 279–300, 2002.
  • S. Favier and F. Zó, “Extension of the best approximation operator in Orlicz spaces and weak-type inequalities,” Abstract and Applied Analysis, vol. 6, no. 2, pp. 101–114, 2001.
  • S. Favier and F. Zó, “A Lebesgue type differentiation theorem for best approximations by constants in Orlicz spaces,” Real Analysis Exchange, vol. 30, no. 1, pp. 29–42, 2005.
  • S. Favier and F. Zó, “Sharp conditions for maximal inequalities čommentPlease update the information of this reference, if possible. of the best approximation operator,” preprint.
  • F. Mazzone and H. Cuenya, “Maximal inequalities and Lebesgue's differentiation theorem for best approximant by constant over balls,” Journal of Approximation Theory, vol. 110, no. 2, pp. 171–179, 2001.
  • H. D. Brunk and S. Johansen, “A generalized Radon-Nikodym derivative,” Pacific Journal of Mathematics, vol. 34, pp. 585–617, 1970.
  • F. Mazzone and H. Cuenya, “A characterization of best $\phi$-approximants with applications to multidimensional isotonic approximation,” Constructive Approximation, vol. 21, no. 2, pp. 207–223, 2005.
  • J. Neveu, Mathematical Foundations of the Calculus of Probability, Holden-Day, San Francisco, Calif, USA, 1965.