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.
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