#### Publication Date

12-2009

#### Abstract

In many practical applications, it is useful to represent a function f(x) by its *fuzzy transform*, i.e., by the "average" values over different elements of a *fuzzy partition* A_{1}(x), ..., _{n}(x) (for which A_{i}(x) >= 0 and A_{1}(x) + ... + A_{n}(x) = 1). It is known that when we increase the number n of the partition elements A_{i}(x), the resulting approximation get closer and closer to the original function: for each value x_{0}, the values F_{i} corresponding to the function A_{i}(x) for which A_{i}(x_{0}) > 0 tend to f(x_{0}).

In some applications, if we approximate the function f(x) on each element A_{i}(x) not by a *constant* but by a *polynomial* (i.e., use a fuzzy transform of a *higher order*), we get an even better approximation to f(x).

In this paper, we show that such fuzzy transforms of higher order (and even sometimes the original fuzzy transforms) not only approximate the function f(x) itself, they also approximate its derivative(s). For example, we have F_{i}'(x_{0}) --> f'(x_{0})$.

## Comments

Technical Report: UTEP-CS-09-39