Minimal multi-convex projections
Tom 178 / 2007
Streszczenie
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 $C^L[0,1]$.
