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 F"i=@!f(x).A"i(x)dx@!A"i(x)dxover different elements of a fuzzy partitionA"1(x),...,A"n(x) (for which A"i(x)>=0 and @?"i"="1^nA"i(x)=1). It is known that when we increase the number n of the partition elements A"i(x), the resulting approximation gets 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)=1 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).