Explicit formulas for C-1,C-1 and C-CONV(1,omega) extensions of 1-jets in Hilbert and superreflexive spaces

摘要

Given X a Hilbert space, omega a modulus of continuity, E an arbitrary subset of X, and functions f : E - R, G : E - X, we provide necessary and sufficient conditions for the jet (f, G) to admit an extension (F, del F) with F : X - R convex and of class C-1,C-omega (X), by means of a simple explicit formula. As a consequence of this result, if omega is linear, we show that a variant of this formula provides explicit C-1,C-1 extensions of general (not necessarily convex) 1-jets satisfying the usual Whitney extension condition, with best possible Lipschitz constants of the gradients of the extensions. Finally, if X is a superreflexive Banach space, we establish similar results for the classes C-CONV(1,alpha)(X). (C) 2017 Elsevier Inc. All rights reserved.