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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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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DOI: https://doi.org/10.1007/978-3-031-46778-3_11
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