Best Interpolation with Convex Constraints

AbstractA characterization of any solution to the minimization problem min{||x − z|| : x ∈ K ≔ C ∩ A−1d} is given, where A is a continuous linear map from a real Banach space X to a locally convex topological space Y, z ∈ X, C ⊂ X is a closed convex set and d ∈ AC. The resulting characterization for the case that X is a Hilbert space is that the projection PK(z) of z to K is PC(z0 + z) for some z0 ∈ ran A* provided d ∈ int AC. An analogous characterization is also obtained for the solution to the nonnegative best interpolation problem in the Lp norm