Minimal multi-convex projections

Abstract

Abstract. We say that a function from X = C L [0, 1] is k-convex (for k ≤ L) if its kth derivative is nonnegative. Let P denote a projection from X onto V = Π n ⊂ X, where Π 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 = (σ 0 , σ 1 , . . . , σ n ) be an (n + 1)-tuple with σ i ∈ {0, 1}; we say f ∈ X is multi-convex if f (i) ≥ 0 for i such that σ i = 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 C L [0, 1].1. Introduction. When X is a Banach space and V ⊂ X a subspace, we denote by P(X, V ) the set of all projections from X onto V ; in the cases where there is no ambiguity, we will simply write P. We say that a projection P 0 is minimal if P 0 ≤ P for all P ∈ P(X, V ).There exist a large number of papers concerning minimal projections. The problems considered are mainly existence ([15] While a minimal projection will, in general, provide good approximations, it may fail to preserve particular properties of elements, as illustrated below. We are therefore motivated to look for projections which leave invari-

Minimal shape-preserving projections in tensor product spaces

“…To accomplish this, we will borrow extensively from [5]. Indeed, to conform to the notation of this paper, letL i andn i denote the positive integers such that L i =L i + m i and n i =n i + m i .…”

“…Indeed, we demonstrate, in Example 4.1, a case in which the tensor product of two minimal shape-preserving projections does not have minimal norm. However, in the specific setting of [5], we can prove the main result from [4] and, consequently, construct minimal shape-preserving projections for tensor product spaces involving C L [0, 1]. In the Sections 2 and 3 we give basic definitions and results from, respectively, tensor product theory and shape-preserving projection theory.…”

“…In the Sections 2 and 3 we give basic definitions and results from, respectively, tensor product theory and shape-preserving projection theory. Section 4 contains the main results, with Example 4.1 demonstrating that the tensor product of two minimal shape-preserving projections need not have minimal norm and Theorem 4.6 accomplishing the goal of generalizing (to tensor product spaces) the results from [5].…”

“…In the paper [5], these two questions were addressed in the case when P is a projection operator (P 2 is the identity map) mapping X = C L [0, 1] (Lth continuously differentiable functions) onto finite-dimensional polynomial spaces. For a large class of cones (or shapes) S, the existence of shape-preserving projections was established and formulas for minimal shapepreserving projections (and their norms) were given.…”

“…For a large class of cones (or shapes) S, the existence of shape-preserving projections was established and formulas for minimal shapepreserving projections (and their norms) were given. The goal of this paper is to extend the results of [5] to tensor product spaces (tensor product spaces are a natural setting in which to construct minimal shape-preserving projections of multi-variate functions; general results in this direction are contained in the (upcoming) paper [6]). One would expect that such a generalization would utilize the main result from [4], which says (roughly speaking) that tensor product of two minimal projections is again a minimal projection (on a tensor product space).…”