Best interpolation in seminorm with convex constraints

Abstract

Implicit and explicit characterizations of the solutions to the following constrained best interpolation problem

$$\min \left\{ {\left\| {Tx - z} \right\|:x \in C \cap A^{ - 1} d} \right\}$$

are presented. Here,T is a densely-defined, closed, linear mapping from a Hilbert spaceX to a Hilbert spaceY, A: X→Z is a continuous, linear mapping withZ a locally, convex linear topological space,C is a closed, convex set in the domain domT ofT, andd∈AC. For the case in whichC is a closed, convex cone, it is shown that the constrained best interpolation problem can generally be solved by finding the saddle points of a saddle function on the whole space, and, if the explicit characterization is applicable, then solving this problem is equivalent to solving an unconstrained minimization problem for a convex function.