Interpolation with function space representation of membership functions

This paper generalizes a previous Cartesian approach for interpolating fuzzy rules comprised of membership functions with finite number of characteristic points. Instead of representing membership functions as points in Cartesian spaces, they now become elements in the space of square, integrable function. Interpolation is thus conducted between the antecedent and consequent function spaces. The generalized representation allows an extended class of membership functions satisfying two monotonicity conditions to be accommodated in the interpolation process. They include the popular bell-shaped membership functions, which were not possible before with the Cartesian representation. The work also extends the similarity triangle-based interpolation technique from the previous Cartesian representation to the new representation. Ensuing issues on computational complexity and nonunique conclusion are discussed. Other concepts such as spanning set and extensibility functions are also presented under the generalized framework. Examples to illustrate the extended approach and to compare with the Cartesian approach are given.