A characterization of best approximations with restricted ranges

Abstract

Restricted range approximation in uniform norm from an extended Haar space of a certain order is an important and widely applicable problem in restricted approximations. In the past 20 years or more, many authors have investigated the characterization of best approximations in various special cases of restricted range approximation, which include approximation with interpolatory constraints, one-sided approximation, and copositive approximation. But the characterization in the general case is still an open question. The paper gives a general characterization theorem in the form of convex hull and alternation. Many known important results are exactly its special cases.