  For an arbitrary subset $E\subset\mathbb{R}^n$ and the scales $\Gamma =
\textit{small, large, far},$ we introduce and study the $\Gamma$-vanishing
subspaces $\dot{\operatorname{VC}}_{\Gamma}^{\omega}(E)\subset
\dot{C}^{0,\omega}(E)$ of the H\"older spaces. We find a single bounded linear
extension operator that maps $\dot{\operatorname{VC}}_{\Gamma}^{\omega}(E)\to
\dot{\operatorname{VC}}_{\Gamma}^{\omega}(\mathbb{R}^n)$ and works separately
for each of the three vanishing scales. In fact, this follows as the zeroth
order special case of a more general problem involving higher order
derivatives. As a corollary, we obtain complete characterizations of
approximability of H\"older functions $\dot{C}^{0,\omega}(E)$ by Lipschitz and
boundedly supported functions, for an arbitrary subset $E\subset\mathbb{R}^n.$
