Extending $n$-convex functions
Volume 171 / 2005
Studia Mathematica 171 (2005), 125-152
MSC: Primary 26A51, 41A05.
DOI: 10.4064/sm171-2-2
Abstract
We are given data $\alpha _1,\mathinner {\ldotp \ldotp \ldotp },\alpha _m$ and a set of points $E=\{ x_1,\mathinner {\ldotp \ldotp \ldotp },x_m\} $. We address the question of conditions ensuring the existence of a function $f$ satisfying the interpolation conditions $f(x_i)=\alpha _i$, $i=1,\mathinner {\ldotp \ldotp \ldotp },m$, that is also $n$-convex on a set properly containing $E$. We consider both one-point extensions of $E$, and extensions to all of ${{\mathbb R}}$. We also determine bounds on the $n$-convex functions satisfying the above interpolation conditions.
