Universal approximation of polygonal fuzzy neural networks in sense of K-integral norms

In this paper, we introduce polygonal fuzzy numbers to overcome the operational complexity of ordinary fuzzy numbers, and obtain two important inequalities by taking advantage of their fine properties. By presenting an actual example, we demonstrate that the approximation capability of polygonal fuzzy numbers is efficient. Furthermore, the concepts of K-quasi-additive integrals and K-integral norms are introduced. Whenever the polygonal fuzzy numbers space satisfies separability, the density problems for several functions spaces can be studied, by means of fuzzy-valued simple functions and fuzzy-valued Bernstein polynomials. We establish that the class of the integrally-bounded fuzzy-valued functions spans a complete and separable metric space in the K-integral norms. Finally, in the sense of K-integral norms, the universal approximation of four-layer regular polygonal fuzzy neural networks for fuzzy-valued simple functions is discussed. Furthermore, we show that this type of networks also possesses universal approximation for the class of integrally-bounded fuzzy-valued functions. This result indicates that the approximation capability which regular polygonal fuzzy neural networks for continuous fuzzy systems can be extended as for general integrable systems.