Skip to main content
Log in

Shape-preserving projections in low-dimensional settings and the q-monotone case

  • Published:
Ukrainian Mathematical Journal Aims and scope

Let P:XV be a projection from a real Banach space X onto a subspace V and let SX. In this setting, one can ask if S is left invariant under P, i.e., if PSS. If V is finite-dimensional and S is a cone with particular structure, then the occurrence of the imbedding PSS can be characterized through a geometric description. This characterization relies heavily on the structure of S, or, more specifically, on the structure of the cone S * dual to S. In this paper, we remove the structural assumptions on S * and characterize the cases where PSS. We note that the (so-called) q-monotone shape forms a cone that (lacks structure and thus) serves as an application for our characterization.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. B. Chalmers, D. Mupasiri, and M. P. Prophet, “A characterization and equations for minimal shape-preserving projections,” J. Approxim. Theory, 138, 184–196 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  2. E. W. Cheney, Introduction to Approximation Theory, Chelsea, New York (1982).

    MATH  Google Scholar 

  3. K. Donner, Extension of Positive Operators and Korovkin Theorems, Springer, Berlin (1982).

    MATH  Google Scholar 

  4. G. Lewicki and M. P. Prophet, “Minimal multi-convex projections,” Stud. Math., 178, No. 2, 99–124 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  5. D. Mupasiri and M. P. Prophet, “A note on the existence of shape-preserving projections,” Rocky Mountain J. Math., 37, No. 2, 573–585 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  6. D. Mupasiri and M. P. Prophet, “On the difficulty of preserving monotonicity via projections and related results,” Jaen J. Approxim., 2, No. 1, 1–12 (2010).

    MathSciNet  MATH  Google Scholar 

  7. H. H. Schaefer, Banach Lattices and Positive Operators, Springer, New York (1974).

    Book  MATH  Google Scholar 

  8. H. Schneider and B. Tam, “On the core of a cone-preserving map,” Trans. Amer. Math. Soc., 343, No. 2, 479–524 (1994).

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 64, No. 5, pp. 674–684, May, 2012.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Prophet, M.P., Shevchuk, I.A. Shape-preserving projections in low-dimensional settings and the q-monotone case. Ukr Math J 64, 767–780 (2012). https://doi.org/10.1007/s11253-012-0677-2

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11253-012-0677-2

Keywords

Navigation