Approximation properties of fuzzy systems for smooth functions and their first-order derivative

The problem of simultaneous approximations of a given function and its derivatives, has been addressed frequently in pure and applied mathematics. In pure mathematics, Bernstein polynomials get their importance from the fact that they provide simultaneous approximation of a function and its derivatives. In neural network theory, feedforward networks were shown to be universal approximators of an unknown function and its derivatives. In this paper, we consider fuzzy logic systems with the membership functions of each input variables are chosen as the translations and dilations of one appropriately fixed function. We prove, by a constructive proof based on discretization of the convolution operator, that under certain conditions made on the input variables membership functions, fuzzy logic systems of Sugeno type are universal approximators of a given function and its derivatives.