A new approach to universal approximation of fuzzy functions on a discrete set of points

One of the interesting, important and attractive problems in applied mathematics is approximation of functions in a given space. In this paper the problem is considered for fuzzy data and fuzzy functions using the defuzzification function of Fortemps and Roubens. Approximation of a fuzzy function on some given points (xi,f˜i) for i = 1, 2, … , m is considered by some researchers as interpolation problem. But in interpolation problem we find a polynomial of degree at most n = m − 1 where m is the number of points. But when we have lots of points (m is very large) it is not good or even possible to find such polynomials. In this case we want to find a polynomial of arbitrary degree which is an approximation to original function. One of the works has done is regression by some researchers and we introduced a different method. In this case we have m points but we want to find a polynomial of degree at most n < m but not n = m − 1 necessarily. We introduce a fuzzy polynomial approximation as universal approximation of a fuzzy function on a discrete set of points and we present a method to compute it. We show that this approximation can be non-unique, however we choose one of them with the smallest amount of fuzziness.