Extending $n$-convex functions

Allan Pinkus¹ ; Dan Wulbert²

¹ Department of Mathematics Technion 32000 Haifa, Israel
² Department of Mathematics University of California, San Diego (UCSD) 9500 Gilman Drive La Jolla, CA 92093-0112, U.S.A.

Studia Mathematica, Tome 171 (2005) no. 2, pp. 125-152

Résumé

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.

Détail

DOI : 10.4064/sm171-2-2

Keywords: given alpha mathinner ldotp ldotp ldotp alpha set points mathinner ldotp ldotp ldotp address question conditions ensuring existence function satisfying interpolation conditions alpha mathinner ldotp ldotp ldotp n convex set properly containing consider one point extensions extensions mathbb determine bounds n convex functions satisfying above interpolation conditions