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Convexity of Functions Produced by Bernstein Operators of Max-Product Kind

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Abstract

Bernstein polynomials have nice mathematical properties and wide applications in various fields. An interesting property which is very useful in computer-aided design is preservation of convexity of functions. These polynomials have been extended in various forms according to different applications. One form considered in this paper is the family of Bernstein operators of max-product kind. These nonlinear operators have many interesting properties including preservation of quasiconvexity of functions, but do not preserve convexity of linear functions. In this paper we strengthen this observation for the linear functions. We show that for any continuous function with nonnegative values, the functions generated by Bernstein operators of max-product kind are not convex unless they are constant functions.

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References

  1. Badea, C.: Bernstein polynomials and operator theory. Results Math. 53, 229–236 (2009)

    Article  MathSciNet  Google Scholar 

  2. Bede, B., Coroianu, L., Gal, S.G.: Approximation and shape preserving properties of the Bernstein operator of max-product kind. Int. J. Math. Math. Sci. 2009, 26 (2009). Article ID 590589

    Article  MathSciNet  Google Scholar 

  3. Bede, B., Coroianu, L., Gal, S.G.: Approximation by Max-Product Type Operators. Springer, Berlin (2016)

    Book  Google Scholar 

  4. Bernstein, S.: Démonstration du théorème de Weierstrass fondée sur le calcul des probabilities. Commun. Soc. Math. Kharkov 13, 1–2 (1912)

    Google Scholar 

  5. Casselman, B.: From Bézier to Bernstein. Feature Column from American Mathematical Society, New York (2008)

    Google Scholar 

  6. Gal, S.G.: Shape-Preserving Approximation by Real and Complex Polynomials. Birkhäuser, Boston (2008)

    Book  Google Scholar 

  7. Gonska, H., Pitul, P., Rasa, I.: General King-type operators. Results Math. 53, 279–286 (2009)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

We would like to thank the editors and referees for their encouraging comments and constructive suggestions.

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Authors and Affiliations

  1. Shaoxing Luxun College (Luxun Gaoji Zhongxue), 367 Cheng Nan Da Dao, Shaoxing, 312000, Zhejiang, People’s Republic of China

    Ruo-Han Shen

  2. Shaoxing University, Shaoxing, 312000, Zhejiang, People’s Republic of China

    Le-Yin Wei

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  1. Ruo-Han Shen
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  2. Le-Yin Wei
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Correspondence to Le-Yin Wei.

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Le-Yin Wei is the corresponding author.

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Shen, RH., Wei, LY. Convexity of Functions Produced by Bernstein Operators of Max-Product Kind. Results Math 74, 92 (2019). https://doi.org/10.1007/s00025-019-1015-x

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  • DOI: https://doi.org/10.1007/s00025-019-1015-x

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