Minimal multi-convex projections

• Autorzy: Grzegorz Lewicki , Michael Prophet
• Języki publikacji: EN

Abstrakty

EN

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 $C^{L}[0,1]$.

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2007
• Tom: 178
• Numer: 2
• Strony: 99-124

Daty

wydano

2007

Twórcy

autor

Grzegorz Lewicki

• Department of Mathematics, Jagiellonian University, Reymonta 4, 30-059 Kraków, Poland

autor

Michael Prophet

• Department of Mathematics, University of Northern Iowa, Cedar Falls, IA 50614-0506, U.S.A.

Identyfikatory

DOI

10.4064/sm178-2-1