For $0<\alpha<1$, membership in $\operatorname{Lip}(\alpha)$ is equivalent, up to constants depending on $\alpha$, to the estimate
\begin{align*}
E_n(f) \lesssim n^{-\alpha}
\end{align*}
for continuous functions on $[-1,1]$.
Knowledge Status
Analysis
Discussion
A theorem in [approximation theory](/page/Approximation%20Theory) concerning Direct-Inverse Principle for Lipschitz Smoothness, used to organise the analytic structure of approximation methods and convergence results.