Minimal multi-convex projections

Grzegorz Lewicki, and Michael Prophet. "Minimal multi-convex projections." Studia Mathematica 178.2 (2007): 99-124. <http://eudml.org/doc/284903>.

@article{GrzegorzLewicki2007,
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 = Πₙ ⊂ X, where Πₙ 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 $σ_\{i\} ∈ \{0, 1\}$; 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 $C^\{L\}[0,1]$.},
author = {Grzegorz Lewicki, Michael Prophet},
journal = {Studia Mathematica},
keywords = {multi-convex functions; minimal; projection; shape preserving projection; generalized representation measure},
language = {eng},
number = {2},
pages = {99-124},
title = {Minimal multi-convex projections},
url = {http://eudml.org/doc/284903},
volume = {178},
year = {2007},
}

TY - JOUR
AU - Grzegorz Lewicki
AU - Michael Prophet
TI - Minimal multi-convex projections
JO - Studia Mathematica
PY - 2007
VL - 178
IS - 2
SP - 99
EP - 124
AB - 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 = Πₙ ⊂ X, where Πₙ 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 $σ_{i} ∈ {0, 1}$; 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 $C^{L}[0,1]$.
LA - eng
KW - multi-convex functions; minimal; projection; shape preserving projection; generalized representation measure
UR - http://eudml.org/doc/284903
ER -