Function approximation is a very important practical problem: in many practical applications, we know the exact form of the functional dependence y=f(x1,...,xn) between physical quantities, but this exact dependence is complicated, so we need a lot of computer space to store it, and a lot of time to process it, i.e., to predict y from the given xi. It is therefore necessary to find a simpler approximate expression g(x1,...,xn) for this same dependence. This problem has been analyzed in numerical mathematics for several centuries, and it is, therefore, one of the most thoroughly analyzed problems of applied mathematics. There are many results related to approximation by polynomials, trigonometric polynomials, splines of different type, etc. Since this problem has been analyzed for so long, no wonder that for many reasonable formulations of the optimality criteria, the corresponding problems of finding the optimal approximations have already been solved.
Lately, however, new clustering-related techniques have been applied to solve this problem (by Yager, Filev, Chu, and others). At first glance, since for most traditional optimality criteria, optimal approximations are already known, clustering approach can only lead to non-optimal approximations, i.e., approximations of inferior quality. We show, however, that there exist new reasonable criteria with respect to which clustering-based function approximation is indeed the optimal method of function approximation.