Some necessary conditions for single-input single-output fuzzy systems as universal approximators

In this paper, we investigate necessary conditions for some typical single-input single-output fuzzy systems as universal approximators for continuous functions defined on compact domains with arbitrarily small uniform approximation error bounds. The necessary conditions we found provide a basis for insightful analysis of strength as well as limitation of the fuzzy systems. The main strength is that only a small number of fuzzy rules may be needed to uniformly approximate continuous functions that have a complicated formulation but a relatively small number of extrema. The limitation is that in order to approximate frequently oscillatory continuous functions, the number of fuzzy rules must be large