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
Anguelov, R.: An Introduction to Some Spaces of Interval Functions, Technical Report UPWT2004/3, University of Pretoria, 2004.
Anguelov, R.: Dedekind Order Completion of C(X) by Hausdorff Continuous Functions, Quaestiones Mathematicae 27 (2004), pp. 153–170.
Anguelov, R. and Markov, S.: Extended Segment Analysis, Freiburger Intervall-Berichte, Inst. Angew. Math, U. Freiburg i. Br. 10 (1981), pp. 1–63.
Anguelov, R. and Minani, F.: Hausdorff Continuous Viscosity Solutions of Hamilton- Jacobi Equations, Journal of Mathematical Analysis and Applications, to appear.
Anguelov, R. and Rosinger, E. E.: Hausdorff Continuous Solutions of Nonlinear PDEs through the Order Completion Method, Quaestiones Mathematicae 28 (2005), pp. 271–285.
Anguelov, R. and van der Walt, J. H.: Order Convergence Structure on C(X), Quaestiones Mathematicae 28 (2005), pp. 425–457.
Artstein, Z.: A Calculus for Set- Valued Maps and Set- Valued Evolution Equations, Set-Valued Analysis 3 (1995), pp. 213–261.
Artstein, Z.: Extensions of Lipschitz Selections and an Application to Differential Inclusions, Nonlinear Analysis: Theory, Methods and Applications 16 (1991), pp. 701–704.
Artstein, Z.: Piecewise Linear Approximations of Set- Valued Maps, Journal of Approximation Theory 56 (1989), pp. 41–47.
Aubin, J. P. and Ekeland, I.: Applied Nonlinear Analysis, Wiley, 1984.
Baire, R.: Lecons sur les Fonctions Discontinues, Collection Borel, Paris, 1905.
Bardi, M. and Capuzzo-Dolcetta, I. : Optimal Control and Viscosity Solutions of Hamilton- Jacobi- Bellman Equations, Birkauser, Boston - Basel - Berlin, 1997.
Birkhoff, G.: The Role of Order in Computing, in: Moore, R. (ed.), Reliability in Computing, Academic Press, 1988, pp. 357–378.
Cellina, A.: On the Differential Inclusion' e (-1, 1), Rend. Ace. Naz. Lincei69 (1980), pp. 1–6.
Clarke, F. H.: Optimization and Nonsmooth Analysis, Wiley-Interscience, 1983.
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.
Dilworth, R. P.: The Normal Completion of the Lattice of Continuous Functions, Trans. Amer. Math. Soc. 68 (1950), pp. 427–438.
Dyn, N. and Farkhi, E.: Spline Subdivision Schemes for Convex Compact Sets, Journal of Computational and Applied Mathematics 119 (2000), pp. 133–144.
Filippov, A. F.: Differential Equations with Discontinuous Right-Hand Side, Nauka, Moskow, 1988. (in Russian).
Ishii, H.: Perron's Method for Hamilton-Jacob Equations, Duke Mathematical Journal 55 (1987), pp. 369–384.
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.
Lempio, F. and Veliov, V.: Discrete Approximation of Differential Inclusions, Bayreuther Math- ematische Schriften 54 (1998), pp. 149–232.
Markov, S.: Extended Interval Arithmetic Involving Infinite Intervals, Mathematica Balkanica 6 (1992), pp. 269–304.
Markov, S.: On Quasilinear Spaces of Convex Bodies and Intervals, J. Comput. Appl. Math. 162 (2004), pp. 93–112.
Rockafellar, R. T.: Convex Analysis, Princeton University Press, 1970.
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.
Sendov, BL: Hausdorff Approximations, Kluwer Academic Publishers, 1990.
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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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