Necessary and sufficient conditions for generalized uniform fuzzy partitions

The fundamental concept in the theory of fuzzy transform (F-transform) is that of fuzzy partition. The original definition assumes that each two fuzzy subsets overlap in such a way that sum of membership degrees in each point is equal to 1. However, this condition can be generalized to obtain a denser fuzzy partition that leads to improvement of approximation properties of Ftransform. However, a problem arises how one can effectively construct such type of fuzzy partitions. We use a generating function having special properties and it is not immediately clear whether it really defines a general uniform fuzzy partition. In this paper, we provide necessary and sufficient condition using which we can solve this task so that optimal generalized uniform fuzzy partition can be designed more easily. This is important in various practical applications of the F-transform, for example in image processing, time series analysis, solving differential equations with boundary conditions, and other ones.