Fuzzy transforms of higher order approximate derivatives: A theorem

In many practical applications, it is useful to represent a function f(x) by its fuzzy transform, i.e., by the “average” values Fi=∫f(x)·Ai(x)dx∫Ai(x)dxover different elements of a fuzzy partition  A1(x),…,An(x)A1(x),…,An(x) (for which Ai(x)≥0Ai(x)≥0 and ∑i=1nAi(x)=1). It is known that when we increase the number n   of the partition elements Ai(x)Ai(x), the resulting approximation gets closer and closer to the original function: for each value x0x0, the values FiFi corresponding to the function Ai(x)Ai(x) for which Ai(x0)=1Ai(x0)=1 tend to f(x0)f(x0). In some applications, if we approximate the function f(x  ) on each element Ai(x)Ai(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 Fi′(x0)→f′(x0).