Faculty Publications

Title

Minimal multi-convex projections

Document Type

Article

Keywords

Minimal projection, Multi-convex function, Shape-preserving projection

Journal/Book/Conference Title

Studia Mathematica

Volume

178

Issue

2

First Page

99

Last Page

124

Abstract

We say that a function from X = CL[0,1] is k-convex (for k ≤ L) if its kth derivative is nonnegative. Let P denote a projection from X onto V =∏n ⊂ X, where ∏n denotes the space of algebraic polynomials of degree less than or equal to n. If we want P to leave invariant the cone of k-convex functions (k ≤ n), we find that such a demand is impossible to fulfill for nearly every k. Indeed, only for k = n-1 and k = n does such a projection exist. So let us consider instead a more general "shape" to preserve. Let σ = (σ0, σ1,...,σn) be an (n + 1)-tuple with σi ∈ {0,1}; we say ∫ ∈ X is multi-convex if f(i) ≥ 0 for i such that σi, = 1. We characterize those σ for which there exists a projection onto V preserving the multi-convex shape. For those shapes able to be preserved via a projection, we construct (in all but one case) a minimal norm multi-convex preserving projection. Out of necessity, we include some results concerning the geometrical structure of CL[0,1].

Original Publication Date

1-1-2007

DOI of published version

10.4064/sm178-2-1

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