The existence of shape-preserving operators with a given action
type: Item
instance of: scholarly article
title: The existence of shape-preserving operators with a given action
zbMATH Open document ID: 0932.47029
author: B. L. Chalmers
author: Michael P. Prophet
published in: Rocky Mountain Journal of Mathematics
publication date: 1999-09-14T00:00:00Z
full work available at URL: cont28-3.html
review text: 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.
Mathematics Subject Classification ID: 47B60
Mathematics Subject Classification ID: 41A10
Mathematics Subject Classification ID: 47N10
zbMATH DE Number: 1309854
zbMATH Keywords: projection
zbMATH Keywords: invariant cone
zbMATH Keywords: shape-preserving operators
zbMATH Keywords: positivity
zbMATH Keywords: monotonicity
zbMATH Keywords: convexity
zbMATH Keywords: action
zbMATH Keywords: dual space
zbMATH Keywords: bipolar theorem
zbMATH Keywords: simplicial cone
MaRDI profile type: Publication
OpenAlex ID: W2025100399
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DOI: 10.1216/RMJM/1181071742
description: scientific article; zbMATH DE number 1309854