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The Set of Hausdorff Continuous Functions— The Largest Linear Space of Interval Functions

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Reliable Computing

Abstract

Hausdorff continuous (H-continuous) functions are special interval-valued functions which are commonly used in practice, e.g. histograms are such functions. However, in order to avoid arithmetic operations with intervals, such functions are traditionally treated by means of corresponding semi-continuous functions, which are real-valued functions. One difficulty in using H-continuous functions is that, if we add two H-continuous functions that have interval values at same argument using point-wise interval arithmetic, then we may obtain as a result an interval function which is not H-continuous. In this work we define addition so that the set of H-continuous functions is closed under this operation. Moreover, the set of H-continuous functions is turned into a linear space. It has been also proved that this space is the largest linear space of interval functions. These results make H-continuous functions an attractive tool in real analysis and provides a bridge between real and interval analysis.

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References

  1. Anguelov, R.: An Introduction to Some Spaces of Interval Functions, Technical Report UPWT2004/3, University of Pretoria, 2004.

  2. Anguelov, R.: Dedekind Order Completion of C(X) by Hausdorff Continuous Functions, Quaestiones Mathematicae 27 (2004), pp. 153–170.

    MATH  MathSciNet  Google Scholar 

  3. Anguelov, R. and Markov, S.: Extended Segment Analysis, Freiburger Intervall-Berichte, Inst. Angew. Math, U. Freiburg i. Br. 10 (1981), pp. 1–63.

    Google Scholar 

  4. Anguelov, R. and Minani, F.: Hausdorff Continuous Viscosity Solutions of Hamilton- Jacobi Equations, Journal of Mathematical Analysis and Applications, to appear.

  5. Anguelov, R. and Rosinger, E. E.: Hausdorff Continuous Solutions of Nonlinear PDEs through the Order Completion Method, Quaestiones Mathematicae 28 (2005), pp. 271–285.

    MATH  MathSciNet  Google Scholar 

  6. Anguelov, R. and van der Walt, J. H.: Order Convergence Structure on C(X), Quaestiones Mathematicae 28 (2005), pp. 425–457.

    MATH  MathSciNet  Google Scholar 

  7. Artstein, Z.: A Calculus for Set- Valued Maps and Set- Valued Evolution Equations, Set-Valued Analysis 3 (1995), pp. 213–261.

    Article  MATH  MathSciNet  Google Scholar 

  8. Artstein, Z.: Extensions of Lipschitz Selections and an Application to Differential Inclusions, Nonlinear Analysis: Theory, Methods and Applications 16 (1991), pp. 701–704.

    MathSciNet  Google Scholar 

  9. Artstein, Z.: Piecewise Linear Approximations of Set- Valued Maps, Journal of Approximation Theory 56 (1989), pp. 41–47.

    Article  MATH  MathSciNet  Google Scholar 

  10. Aubin, J. P. and Ekeland, I.: Applied Nonlinear Analysis, Wiley, 1984.

  11. Baire, R.: Lecons sur les Fonctions Discontinues, Collection Borel, Paris, 1905.

    MATH  Google Scholar 

  12. Bardi, M. and Capuzzo-Dolcetta, I. : Optimal Control and Viscosity Solutions of Hamilton- Jacobi- Bellman Equations, Birkauser, Boston - Basel - Berlin, 1997.

    MATH  Google Scholar 

  13. Birkhoff, G.: The Role of Order in Computing, in: Moore, R. (ed.), Reliability in Computing, Academic Press, 1988, pp. 357–378.

  14. Cellina, A.: On the Differential Inclusion' e (-1, 1), Rend. Ace. Naz. Lincei69 (1980), pp. 1–6.

    MathSciNet  Google Scholar 

  15. Clarke, F. H.: Optimization and Nonsmooth Analysis, Wiley-Interscience, 1983.

  16. Crandal, M. G., Ishii, H., and Lions, P.-L.: User's Guide to Viscosity Solutions of Second Order Partial Differential Equations, Bulletin of the American Mathematical Society 27 (1992), pp. 1–67.

    Article  Google Scholar 

  17. Dilworth, R. P.: The Normal Completion of the Lattice of Continuous Functions, Trans. Amer. Math. Soc. 68 (1950), pp. 427–438.

    Article  MATH  MathSciNet  Google Scholar 

  18. Dyn, N. and Farkhi, E.: Spline Subdivision Schemes for Convex Compact Sets, Journal of Computational and Applied Mathematics 119 (2000), pp. 133–144.

    Article  MATH  MathSciNet  Google Scholar 

  19. Filippov, A. F.: Differential Equations with Discontinuous Right-Hand Side, Nauka, Moskow, 1988. (in Russian).

    Google Scholar 

  20. Ishii, H.: Perron's Method for Hamilton-Jacob Equations, Duke Mathematical Journal 55 (1987), pp. 369–384.

    Article  MATH  MathSciNet  Google Scholar 

  21. Korovkin, P. P.: An Attempt for Axiomatic Treatment of Some Problems of Approximation Theory, in: Sendov, Bl. (ed.), Constructive Function Theory, BAS, Sofia, 1972, pp. 55–63.

    Google Scholar 

  22. Lempio, F. and Veliov, V.: Discrete Approximation of Differential Inclusions, Bayreuther Math- ematische Schriften 54 (1998), pp. 149–232.

    MATH  MathSciNet  Google Scholar 

  23. Markov, S.: Extended Interval Arithmetic Involving Infinite Intervals, Mathematica Balkanica 6 (1992), pp. 269–304.

    MATH  MathSciNet  Google Scholar 

  24. Markov, S.: On Quasilinear Spaces of Convex Bodies and Intervals, J. Comput. Appl. Math. 162 (2004), pp. 93–112.

    Article  MATH  MathSciNet  Google Scholar 

  25. Rockafellar, R. T.: Convex Analysis, Princeton University Press, 1970.

  26. Sendov, BL: Approximation of Functions by Algebraic Polynomials with Respect to a Metric of Hausdorff Type, Ann. Sofia University, Mathematics 55 (1962), pp. 1–39.

    MathSciNet  Google Scholar 

  27. Sendov, BL: Hausdorff Approximations, Kluwer Academic Publishers, 1990.

  28. Sendov, BL, Anguelov, R., and Markov, S.: On the Linear Space of Hausdorff Continuous Functions, lecture delivered at the llth GAMM-IMACS International Symposium on Scientific Computing, Computer Arithmetic, and Validated Numerics, 4–8 October 2004, Fukuoka, Japan.

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Authors and Affiliations

  1. Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria, South Africa

    Roumen Anguelov

  2. Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, “Acad. G. Bonchev” st., block 8, 1113, Sofia, Bulgaria

    Svetoslav Markov

  3. Bulgarian Embassy, 36-3, Yoyogi 5-chome, Shibuia-ku, Tokyo, 151-0053, Japan

    Blagovest Sendov

Authors
  1. Roumen Anguelov
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  2. Svetoslav Markov
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  3. Blagovest Sendov
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Correspondence to Roumen Anguelov.

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Anguelov, R., Markov, S. & Sendov, B. The Set of Hausdorff Continuous Functions— The Largest Linear Space of Interval Functions. Reliable Comput 12, 337–363 (2006). https://doi.org/10.1007/s11155-006-9006-5

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  • DOI: https://doi.org/10.1007/s11155-006-9006-5

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