A HAAR-TYPE THEORY OF BEST Lx -APPROXIMATION WITH CONSTRAINTS

A general setting for constrained Z,1-approximation is presented. Let Un be a finite dimensional subspace of C[a, b] and L be a linear operator from Un to C(K) (r = 0, 1) where K is a finite union of disjoint, closed, bounded intervals. For v , u e C(K) with v < u, the approximating set is Univ, u) = {p e Un : v < Lp < u on K} and the norm is \\f\\w = Xf \f\wdx where w a positive continuous function on [a, b]. We obtain necessary and sufficient conditions for Un (v, u) to admit unique best || • ||w-approximations to all / € C[a, b] for all positive continuous w and all v , u € C(K) (r = 0, 1) satisfying a nonempty interior condition. These results are applied to several L1-approximation problems including polynomial and spline approximation with restricted derivatives, lacunary polynomial approximation with restricted derivatives, and others.