Fuzzy systems approximation by frames-SISO case

Abstract

In this paper, the approximation problem of SISO fuzzy systems by frames is discussed. Based on the fact that fuzzy systems can be represented by a linear combination of fuzzy basis functions (FBF), we first discuss the mathematical theory of frames in a Hilbert space, and then the conditions under which a FBF is a valid frame. Fuzzy membership functions which constitute a valid frame can universally approximate any continuous function. The triangular membership function is demonstrated as such, and it's approximation properties are discussed.<>