Minimal multi-convex projections

Grzegorz Lewicki¹ ; Michael Prophet²

¹ Department of Mathematics Jagiellonian University Reymonta 4 30-059 Kraków, Poland
² Department of Mathematics University of Northern Iowa Cedar Falls, IA 50614-0506, U.S.A.

Studia Mathematica, Tome 178 (2007) no. 2, pp. 99-124

Résumé

We say that a function from $X=C^L[0,1]$ is $k$-convex (for $k \leq L$) if its $k$th derivative is nonnegative. Let $P$ denote a projection from $X$ onto $V=\varPi_n \subset X$, where $\varPi_n$ 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 ($k \leq n$), 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 $\sigma=( \sigma_0, \sigma_1, \dots, \sigma_n)$ be an $(n+1)$-tuple with $\sigma_i \in \{0, 1 \}$; we say $f \in X$ is multi-convex if $f^{(i)} \geq 0$ for $i$ such that $\sigma_i=1$. We characterize those $\sigma$ 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 $C^L[0,1]$.

Détail

DOI : 10.4064/sm178-2-1

Keywords: say function k convex leq its kth derivative nonnegative denote projection varpi subset where varpi denotes space algebraic polynomials degree equal want leave invariant cone k convex functions leq demand impossible fulfill nearly every indeed only n does projection exist consider instead general shape preserve sigma sigma sigma dots sigma tuple sigma say multi convex geq sigma characterize those sigma which there exists projection preserving multi convex shape those shapes able preserved via projection construct minimal norm multi convex preserving projection out necessity include results concerning geometrical structure