The choice of fuzzy set function affects how well the fuzzy systems approximate an unknown function. We first examine several popular fuzzy systems, based on which we generalize the error of the fuzzy system by the mathematical functional. Two classes of extended fuzzy systems with special set function are proposed respectively in one-dimensional and two-dimensional reproducing kernel spaces, and they are in fact the best interpolation operators in the sense of classical mathematical approximation. The extended fuzzy system can be used as easily as any typical fuzzy system but has better characters. Our major contributions are as follows.

1.   We prove that the existing fuzzy system with a triangle-shaped set function is by no means the best choice for an uncertain system. On the contrary, the reproducing kernel-shaped set function can give the best approximation in the sense of mathematical functional. According to the result, overlapped symmetric triangles or trapezoid set functions reduce fuzzy systems to piecewise linear systems. Universal Gaussian bell-curve set functions can work well but its learning cost is too expensive to apply under online condition.

2.   We explain what our proposed extended fuzzy system is useful for a given uncertain system. This solution is associated with another problem: Is the reproducing kernel condition too specific to limit its applications? We must stress that the reproducing kernel function is easily found in any continuous function space. Consequently, the extended fuzzy system can approximate any continuous function with arbitrary accuracy.

Furthermore one may question why we do not directly use an optimal approximation functional instead of the extended fuzzy system. Here we answer this problem by the following point of views: The two methods face different applicable conditions. The optimal approximation functional is used to interpolate those accurate data. These data are needed under certain conditions such as distributions and quantity, etc. When little information is available, an optimal approximation functional will do nothing. But the extended fuzzy system gives a good approximation to an uncertain system and can work efficiently under the condition of vague or poor information. Consequently, the two methods have different optimal criteria and different input and output descriptions. Furthermore, the extended fuzzy systems can be created flexibly by artificial experiences or other ways. Two experiments are used to verify the effectiveness and efficiency of our proposed extended fuzzy systems.