Best mean approximation by splines satisfying generalized convexity constraints

A characterization of the best L₁-approximation to a continuous function by classes of fixed-knot polynomial splines which satisfy generalized convexity constraints is presented and uniqueness is shown. Included is the possibility of specifying the positivity, monotonicity, or convexity of the class. The proof of uniqueness uses recently developed results for Hermite-Birkhoff interpolation by splines.