Some results on best L1-approximation of continuous functions

The chief purpose of this paper is to study the question of uniqueness of best L¹-approximations for weighted approximation of continuous vector-valued functions and continuous functions of several variables by finite dimensional subspaces G. A uniqueness result for special subspaces G of the type G=G¹, ×…×G^m is given. Moreover It is studied whether a sufficient condition for uniqueness introduced by DeVore and Strauss and in a more general context by Kroó is also necessary or not and a class of finite dimensional subspaces G for which a positive answer can be given is presented. Finally it is shown that subspaces of linear spline functions of two variables fail to have that sufficient property.