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Hausdorff Continuous Interval Functions and Approximations

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Scientific Computing, Computer Arithmetic, and Validated Numerics (SCAN 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9553))

Abstract

The set of interval Hausdorff continuous functions constitutes the largest space preserving basic algebraic and topological structural properties of continuous functions, such as linearity, ring structure, Dedekind order completeness, etc. Spaces of interval functions have important applications not only in the construction of numerical methods and algorithms, but to problems in abstract areas such as real analysis, set-valued analysis, approximation theory and the analysis of PDEs. In this work, we summarize some basic results about the family of interval Hausdorff continuous functions that make interval analysis a bridge between numerical and real analysis. We focus on some approximation issues formulating a new result on the Hausdorff approximation of Hausdorff continuous functions by interval step functions. The Hausdorff approximation of the Heaviside interval step function by sigmoid functions arising from biological applications is also considered, and an estimate for the Hausdorff distance is obtained.

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Notes

  1. 1.

    For brevity we shall further write “interval function” instead of “interval-valued function”.

  2. 2.

    In topology, a meagre set, also called a set of first Baire category, is a set that, considered as a subset of a (usually larger) topological space, is small or negligible.

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Acknowledgments

RA acknowledges partial support of the National Research Foundation of South Africa. RA and SM acknowledge partial support by the Institute of Mathematics and Informatics at the Bulgarian Academy of Sciences. The authors thank Prof. Kamen Ivanov for the analysis and derivation of formula (9). They are grateful to the anonimous reviewer for his careful reading and many remarks.

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Authors and Affiliations

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

    Roumen Anguelov

  2. Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Sofia, Bulgaria

    Roumen Anguelov & Svetoslav Markov

Authors
  1. Roumen Anguelov
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  2. Svetoslav Markov
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Correspondence to Svetoslav Markov .

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Editors and Affiliations

  1. Institute of Computer Science, University of Würzburg, Würzburg, Germany

    Marco Nehmeier

  2. Universität Würzburg, Würzburg, Germany

    Jürgen Wolff von Gudenberg

  3. Department of Mathematics, Uppsala University, Uppsala, Sweden

    Warwick Tucker

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Anguelov, R., Markov, S. (2016). Hausdorff Continuous Interval Functions and Approximations. In: Nehmeier, M., Wolff von Gudenberg, J., Tucker, W. (eds) Scientific Computing, Computer Arithmetic, and Validated Numerics. SCAN 2015. Lecture Notes in Computer Science(), vol 9553. Springer, Cham. https://doi.org/10.1007/978-3-319-31769-4_1

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  • DOI: https://doi.org/10.1007/978-3-319-31769-4_1

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