Necessary and sufficient conditions for generalized uniform fuzzy partitions

The fundamental concept in the theory of fuzzy transform (F-transform) is the fuzzy partition, which is a generalization of the classical concept of partition. The original definition assumes that every two normal fuzzy subsets in a partition overlap in such a way that the sum of the membership degrees at each point is equal to 1. This condition can be generalized by relaxing the assumption of normality for fuzzy sets. The result is a denser fuzzy partition that may improve the approximation properties and the smoothness of the inverse F-transform. A fuzzy partition with this property will be referred to as general. The problem is how a general fuzzy partition can be effectively constructed. If we use a generating function with special properties, then it is not immediately clear whether it defines a general fuzzy partition. In this paper, we find a necessary and sufficient condition that will enable the optimal generalized fuzzy partition to be designed more easily, which is important in various practical applications of the F-transform, for example, image processing, time series analysis, and solving differential equations with boundary conditions.