Applications of a separation principle in good and best approximation theory

Abstract

Most results of best approximation theory are known to be intimately related to various forms of the Hahn-Banach extension theorem. The present paper attempts to highlight: how an elementary geometrical separation principle pervades much of the good and best approximation theory, by showing how all characterizations old and new are indeed potential variants of this principle. The main result consists in a necessary and sufficient characterization condition for good approximations in arbitrary subsets. Moreover a striking result is obtained simply by particularizing this main theorem to best nonlinear approximation theory. Indeed a necessary and sufficient characterization condition is formulated which is shown to be a generalization of the so-called Global Kolmogoroff condition.