Traces of vanishing H\"older spaces

For an arbitrary subset $E\subset\mathbb{R}^n,$ we introduce and study the three vanishing subspaces of the H\"older space $\dot{C}^{0,\omega}(E)$ consisting of those functions for which the ratio $|f(x)-f(y)|/\omega(|x-y|)$ vanishes, when $(1)$ $|x-y|\to 0$ , $(2)$ $|x-y|\to\infty$ or $(3)$ $\min(|x|,|y|)\to\infty.$ We prove that the Whitney extension operator maps each of these vanishing subspaces from $E$ to the corresponding vanishing spaces defined on the whole ambient space $\mathbb{R}^n.$ In fact, this follows as the zeroth order special case of a more general problem involving higher order derivatives. As a consequence, we obtain complete characterizations of approximability of H\"older functions $\dot{C}^{0,\omega}(E)$ by Lipschitz and boundedly supported functions.