The existence of shape-preserving operators with a given action (Q1293500)

In approximation theory, one often wants to replace a function by a simpler function, while preserving some shape (e.g. positivity, monotonicity, convexity) that it may possess. Translated into functional analysis, this means finding a linear operator from a Banach space onto a finite dimensional subspace which leaves a closed convex cone invariant. This is not always possible: the second degree polynomials and monotonicity provide a fundamental counterexample. Accordingly the authors consider a more general problem: given an action (i.e. an operator from the subspace into itself, not necessarily the identity), find an extension to the whole space which leaves the cone invariant. As often happens, the problem is more easily tackled in the dual space. A necessary and sufficient condition is given for the existence of such an extension operator, in terms of finite dimensional subcones of the dual cone. (Another application of functional analysis: Lemma 1.1 could have been proved more easily by noting the bipolar theorem.) Stronger results are obtained for simplicial cones. There are many illustrative examples.