Let $f \in \mathrm{Lip}^{L,\alpha}[a,b]$, and let $f_{h} \in C^{L}$ be a spline which interpolates $f$ at $a + ih$. Then [standard theorems](https://epubs.siam.org/doi/pdf/10.1137/0522089) show that
\begin{align*}
\left\| f - f_{h} \right\|_{\infty} \le Ch^{(\alpha+L)}
\end{align*}
I'm interested in if there are any theorems which bound the error from below, i.e., does there exist $C_1 > 0$ such that
\begin{align*}
\left\| f - f_{h} \right\|_{\infty} \ge  C_1h^{(\alpha+L)}?
\end{align*}
Obviously, if $f$ is a polynomial of degree $L$, no such constant exists, so we first must assume $f \in \mathrm{Lip}^{L,\alpha}[a,b]$, but *not* in $f \in \mathrm{Lip}^{L,\alpha + \epsilon}[a,b]$. Second, the set of points at which $f$ is not infinitely differentiable must be infinite, or else we can just divide $[a,b]$ into subintervals where everything converges faster.

Essentially, I want to know if the Holder regularity of $f$ somehow fundamentally limits how fast we can compute approximations to $f$. (For example, the same sort of convergence rates hold for trigonometric polynomial approximations as well; this is Jackson's theorem.)