Convex C1 extensions of 1-jets from compact subsets of Hilbert spaces

被引:0
作者
Azagra, Daniel [1 ]
Mudarra, Carlos [2 ]
机构
[1] Univ Complutense, Fac Ciencias Matemat, Dept Anal Matemat & Matemat Aplicada, ICMAT CSIC UAM UC3 UCM, Madrid 28040, Spain
[2] Aalto Univ, Dept Math & Syst Anal, POB 11100, FI-00076 Aalto, Finland
基金
芬兰科学院;
关键词
D O I
10.5802/crmath.62
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
Let X denote a Hilbert space. Given a compact subset K of X and two continuous functions f : K -> R., G : K -> X, we show that a necessary and sufficient condition for the existence of a convex function F epsilon C-1 (X) such that F = f on K and del F = G on K is that the 1-jet (f, G) satisfies: (1) f(x) >= f(y) + < G(y), x - y > for all x, y epsilon K, and (2) if x, y epsilon K and f(x) = f(y) + < G(y), x - y > then G(x) =G(y). We also solve a similar problem for K replaced with an arbitrary bounded subset of X, and for C-1 (X) replaced with the class C-b(1,u) (X) of differentiable functions with uniformly continuous derivatives on bounded subsets of X.
引用
收藏
页码:551 / 556
页数:6
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