Approximation properties of piece-wise parabolic functions fuzzy logic systems

Spline functions are widely used in the field of approximation theory and its applications. In this article, we propose the use of π functions which are spline functions of degree two, as membership functions of the input variables. At first, we show by a constructive theory that multi-input-single-output (MISO) Sugeno Fuzzy Logic Systems with π input variables membership functions are universal approximators to every continuously differentiable function defined on a compact subset of Rn. Moreover they are of first order approximation accuracy. Then, we show that with appropriately chosen parameters of the π functions, fuzzy logic systems with π input variables membership functions are universal approximators not only for the function itself but also for its first order derivative.