Extremal problems of approximation theory in fuzzy context

The problem of approximation of a fuzzy subset of a normed space is considered. We study the error of approximation, which in this case is characterized by an L-fuzzy number. In order to do this we define the supremum of an L-fuzzy set of real numbers as well as the supremum and the infimum of a crisp set of L-fuzzy numbers. The introduced concepts allow us to investigate the best approximation and the optimal linear approximation. In particular, we consider approximation of a fuzzy subset in the space L^m_p of differentiable functions in the L_q-metric. We prove the fuzzy counterparts of duality theorems, which in crisp case allows effectively to solve extremal problems of the classical approximation theory.