STABILITY AND CONVERGENCE OF EXTENSION SCHEMES TO CONTINUOUS-FUNCTIONS IN GENERAL METRIC-SPACES

For any E, E' general metric spaces, we formulate the concept of stabi lity of an extension scheme epsilon (phi continuous mapping from some closed subset of E into E', epsilon(phi) continuous and extending phi) . We show that, when E' = IR, stable extension schemes always exist an d that the classical extension schemes in the literature are instable. We also show that, when E' is complete, any stable extrapolation sche me epsilon (phi mapping from some discrete and closed subset of E into E', epsilon(phi) continuous and extending phi) has a unique extension to a stable extension scheme: this result establishes a link between the problem of extrapolation, which usually refers to numerical analys is, and the problem of extension, which also concerns pure mathematics .