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Theoretical Explanation of Bernstein Polynomials’ Efficiency

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Fuzzy Information Processing 2023 (NAFIPS 2023)

Part of the book series: Lecture Notes in Networks and Systems ((LNNS,volume 751))

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Abstract

Many fuzzy data processing problems can be reduced to problems of interval computations. In many applications of interval computations, it turned out to be beneficial to represent polynomials on a given interval \([\underline{x},\overline{x}]\) as linear combinations of Bernstein polynomials \((x-\underline{x})^k\cdot (\overline{x}-x)^{n-k}\). In this paper, we provide a theoretical explanation for this empirical success: namely, we show that under reasonable optimality criteria, Bernstein polynomials can be uniquely determined from the requirement that they are optimal combinations of optimal polynomials corresponding to the interval’s endpoints.

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Acknowledgments

This work was supported in part by the National Science Foundation grants 1623190 (A Model of Change for Preparing a New Generation for Professional Practice in Computer Science), and HRD-1834620 and HRD-2034030 (CAHSI Includes), and by the AT &T Fellowship in Information Technology. It was also supported by the program of the development of the Scientific-Educational Mathematical Center of Volga Federal District No. 075-02-2020-1478, and by a grant from the Hungarian National Research, Development and Innovation Office (NRDI).

The author is thankful to the anonymous referees for valuable suggestions.

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Authors and Affiliations

  1. Department of Computer Science, University of Texas at El Paso, 500 W. University, El Paso, TX, 79968, USA

    Vladik Kreinovich

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  1. Vladik Kreinovich
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Correspondence to Vladik Kreinovich .

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Editors and Affiliations

  1. University of Cincinnati, Cincinnati, OH, USA

    Kelly Cohen

  2. Thales Avionics, Cincinnati, OH, USA

    Nicholas Ernest

  3. DigiPen Institute of Technology, Redmond, WA, USA

    Barnabas Bede

  4. Department of Computer Science, University of Texas at El Paso, El Paso, TX, USA

    Vladik Kreinovich

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Kreinovich, V. (2023). Theoretical Explanation of Bernstein Polynomials’ Efficiency. In: Cohen, K., Ernest, N., Bede, B., Kreinovich, V. (eds) Fuzzy Information Processing 2023. NAFIPS 2023. Lecture Notes in Networks and Systems, vol 751. Springer, Cham. https://doi.org/10.1007/978-3-031-46778-3_11

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