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Multivariate fuzzy transform of complex-valued functions determined by monomial basis

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Abstract

In this paper, we introduce the multivariate fuzzy transform of higher degree of complex-valued functions. Apart from the orthogonal bases of multivariate complex polynomials of weighted Hilbert spaces that are derived by the Gram–Schmidt orthogonalization process, which can be problematic and imprecise in certain cases, we propose to compute the multivariate fuzzy transform components using a simple matrix calculus with the help of the monomial bases. By this novel approach, we derive two types of upper bound of the approximation error both of multivariate complex-valued functions and of their partial derivatives (the latter by the multivariate higher degree fuzzy transform). The results are demonstrated on examples.

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Notes

  1. It means that the real part as well as the imaginary part is bounded piecewise continuous real-valued n-variate functions.

  2. Note that a fair comparison of methods is beyond of the scope of this paper, because a suitable choice of the parameters of the respective method can significantly influence the quality of particular approximation. A preliminary comparison of methods with the univariate higher degree F-transform can be found in Holčapek et al. (2016).

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Acknowledgements

This work was supported by the project LQ1602 IT4Innovations excellence in science. The additional support was provided by the Czech Science Foundation through the Project of No.16-09541S.

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Authors and Affiliations

  1. Institute for Research and Applications of Fuzzy Modeling, CE IT4Innovations, University of Ostrava, 30. dubna 22, 701 03, Ostrava, Czech Republic

    Linh Nguyen, Michal Holčapek & Vilém Novák

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  1. Linh Nguyen
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  2. Michal Holčapek
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  3. Vilém Novák
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Correspondence to Michal Holčapek.

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Communicated by F. Di Martino, V. Novák.

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Nguyen, L., Holčapek, M. & Novák, V. Multivariate fuzzy transform of complex-valued functions determined by monomial basis. Soft Comput 21, 3641–3658 (2017). https://doi.org/10.1007/s00500-017-2658-8

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