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Higher Degree F-transforms Based on B-splines of Two Variables

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Information Processing and Management of Uncertainty in Knowledge-Based Systems (IPMU 2016)

Part of the book series: Communications in Computer and Information Science ((CCIS,volume 610))

Abstract

The paper deals with the higher degree fuzzy transforms (F-transforms with polynomial components) for functions of two variables in the case when two-dimensional generalized fuzzy partition is given by B-splines of two variables. We investigate properties of the direct and inverse F-transform in this case and prove that using B-splines as basic functions of fuzzy partition allows us to improve the quality of approximation.

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Notes

  1. 1.

    In fact, this way a semi-norm \( \left\| \cdot \right\| _{i,j} \) in \( L_2(A_i \times B_j) \) is defined. The identification of functions f and g, which are almost everywhere equal in the sense \( \left\| f-g \right\| _{i,j} = 0 \), turns \( \left\| \cdot \right\| _{i,j} \) from a semi-norm into a norm and the semi-definite sesquilinear form \( \left\langle \cdot ,\cdot \right\rangle _{i,j} \) into an inner product. Further on, this technical detail will be ignored.

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Latvia, Zellu 25, Riga, 1002, Latvia

    Martins Kokainis & Svetlana Asmuss

  2. Institute of Mathematics and Computer Science, University of Latvia, Raina Bulvaris 29, Riga, 1459, Latvia

    Svetlana Asmuss

Authors
  1. Martins Kokainis
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  2. Svetlana Asmuss
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Corresponding author

Correspondence to Martins Kokainis .

Editor information

Editors and Affiliations

  1. INESC-ID,Instituto Superior Técnico, Universidade de Lisboa, Lisboa, Portugal

    Joao Paulo Carvalho

  2. LIP 6, Université Pierre et Marie Curie, Paris, France

    Marie-Jeanne Lesot

  3. School of Industrial Engineering, Eindhoven University of Technology, Eindhoven, The Netherlands

    Uzay Kaymak

  4. IDMEC,Instituto Superior Técnico, Universidade de Lisboa, Lisboa, Portugal

    Susana Vieira

  5. LIP6, Université Pierre et Marie Curie, CNRS, Paris, France

    Bernadette Bouchon-Meunier

  6. Machine Intelligence Institute, Iona College, New Rochelle, New York, USA

    Ronald R. Yager

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Kokainis, M., Asmuss, S. (2016). Higher Degree F-transforms Based on B-splines of Two Variables. In: Carvalho, J., Lesot, MJ., Kaymak, U., Vieira, S., Bouchon-Meunier, B., Yager, R. (eds) Information Processing and Management of Uncertainty in Knowledge-Based Systems. IPMU 2016. Communications in Computer and Information Science, vol 610. Springer, Cham. https://doi.org/10.1007/978-3-319-40596-4_54

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  • DOI: https://doi.org/10.1007/978-3-319-40596-4_54

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-40595-7

  • Online ISBN: 978-3-319-40596-4

  • eBook Packages: Computer ScienceComputer Science (R0)

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