Paper:

# Max-Product Shepard Approximation Operators

## Barnabás Bede^{*}, Hajime Nobuhara^{**}, János Fodor^{***},

and Kaoru Hirota^{**}

^{*}Department of Mechanical and System Engineering, Bánki Donát Faculty of Mechanical Engineering, Budapest Tech, Népszinház u. 8, H-1081 Budapest, Hungary

^{**}Department of Computational Intelligence and Systems Science, Tokyo Institute of Technology, G3-49, 4259 Nagatsuta, Midoriku, Yokohama 226-8502, Japan

^{***}Institute of Intelligent Engineering Systems, John von Neumann Faculty of Informatics, Budapest Tech, Bécsi út 96/b, H-1034 Budapest, Hungary

In crisp approximation theory the operations that are used are only the usual sum and product of reals. We propose the following problem: are sum and product the only operations that can be used in approximation theory? As an answer to this problem we propose max-product Shepard approximation operators and we prove that these operators have very similar properties to those provided by the crisp approximation theory. In this sense we obtain uniform approximation theorem of Weierstrass type, and Jackson-type error estimate in approximation by these operators.

and Kaoru Hirota, “Max-Product Shepard Approximation Operators,”

*J. Adv. Comput. Intell. Intell. Inform.*, Vol.10, No.4, pp. 494-497, 2006.

- [1] G. A. Anastassiou, “Rate of convergence of fuzzy neural network operators, univariate case,” J. Fuzzy Math. Vol.10, No.3, pp. 755-780, 2002.
- [2] G. A. Anastassiou, and S. G. Gal, “Approximation Theory: Moduli of Continuity and Global Smoothness Preservation,” Birkhäuser, Boston-Basel-Berlin, 2000.
- [3] R. A. Devore, and G. G. Lorentz, “Constructive Approximation, Polynomials and Splines Approximation,” Springer-Verlag, Berlin, Heidelberg, 1993.
- [4] A. DiNola, S. Sessa, W. Pedrycz, and E. Sanchez, “Fuzzy Relation Equation and Their Applications to Knowledge Engineering,” Kluwer Academic Publishers, 1989.
- [5] G. Ferrari-Trecate, and R. Rovatti, “Fuzzy systems with overlapping Gaussian concepts: Approximation properties in Sobolev norms,” Fuzzy Sets and Systems, 130, pp. 137-145, 2002.
- [6] L. Horváth, and I. J. Rudas, “Modeling and Problem Solving Methods for Engineers,” ISBN 0-12-602250-X, Elsevier, Academic Press, 2004.
- [7] L. T. Kóczy, and K. Hirota, “Approximate reasoning by linear rule interpolation and general approximation,” International Journal of Approximate Reasoning, 9, pp. 197-225, 1993.
- [8] P. Liu, “Universal approximations of continuous fuzzy-valued functions by multi-layer regular fuzzy neural networks,” Fuzzy Sets and Systems, 119, pp. 313-320, 2001.
- [9] E. H. Mamdani, and S. Assilian, “An experiment in linguistic synthesis with a fuzzy logic controller,” J. Man Machine Stud., 7, pp. 1-13, 1975.
- [10] E. Pap, and K. Jegdić, “Pseudo-analysis and its application in railway rooting,” Fuzzy Sets and Systems, 116, pp. 103-118, 2000.
- [11] J. Szabados, “On a problem of R. DeVore,” Acta Math. Hungar., 27(1-2), pp. 219-223, 1976.
- [12] D. Tikk, “Notes on the approximation rate of fuzzy KH interpolators,” Fuzzy Sets and Systems, 138, pp. 441-453, 2003.