Polytechnic University of Valencia Congress, Workshop Applied Topological Structures WATS'17

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A version of Stone-Weierstrass theorem in Fuzzy Analysis
Juan J. Font, Delia Sanchis, Manuel Sanchis

Last modified: 01-10-2019

Abstract


Fuzzy numbers provide formalized tools to deal with non-precise quantities. They are indeed fuzzy sets in the real line and were introduced in 1978 by Dubois and Prade , who also defined their basic operations. Since then, Fuzzy Analysis has developed based on the notion of fuzzy number just as much as classical Real Analysis did based on the concept of real number. Such development was eased by a characterization of fuzzy numbers provided in 1986 by Goetschel and Voxman leaning on their level sets.

As in the classical setting, continuous fuzzy-valued functions (fuzzy functions) are the central core of the theory. The principal difference with regard to real-valued continuous functions is the fact that the fuzzy numbers do not form a vectorial space, which determines all the results, and, especially, the proofs. The study of fuzzy functions has developed, principally, about two lines of investigation:

- Differential fuzzy equations, which have turned out to be the natural way of modelling physical and engineering problems in contexts where the parameters are vague or incomplete.

- The problem of approximation of fuzzy functions, basically using the approximation capability of fuzzy neural networks.

 

We will focus on this second line of investigation, though our approach will be more general and based on an adaptation of the famous Stone-Weierstrass Theorem to the fuzzy context. This way so, we introduce the concept of “multiplier” of a set of fuzzy functions and use it to give a constructive proof of a Stone-Weiestrass type theorem for fuzzy functions.


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