General characterization of best approximation and its application to uniform approximation with restricted ranges

Let R be a normed linear space, K be an arbitrary convex subset of an n-dimensional subspace Φ _n ⊂R. This paper first gives a general charactaerization for a best approximation from K in form of “zero in the convex hull”. Applying it to the uniform approximation by generalized polynomials with restricted ranges, we get further an alternation characterization. Our results ocntains the special cases of interpolatory approximation, positive approximation, copositive approximation, and the classical characterizations in forms of convex hull and alternation in approximation without restriction.