Dual cones, constrained \(n\)-convex, \(L_ p\)-approximation, and perfect splines (Q1890563)

The authors study constrained \(L_ p\)-approximation, where the approximating set is a convex subcone of \(n\)-convex functions determined by certain constraints. The existence of a best \(L_ p\)-approximation and its characterization is established by determining first a generating basis and then the dual cone of the approximating subcone. Two special cases are studied in detail: \(L_ 2\)-approximation by nondecreasing functions, and the characterization of a best \(L_ 1\)-approximation to a continuous function from the subcone in terms of perfect splines.