A. Shadrin (Acta Mathematica, 2001) shows that the $L_\infty$ norm of the $L_2$ projector $P_\Delta$ onto the spline space $S_k(\Delta$) is bounded independently of the knot-sequence. I.e. for a constant $C_k$ that only depends on the order of the spline space (not $\Delta$)
$$ \| P_{\Delta}(f )\|_\infty \leq C_k \| f\|_\infty.$$
Hall & Meyer (J. Approx. Theory, 1976) show for $f \in C^4[a,b]$ and a mesh $a = x_1, \ldots x_n = b$ with $h = \max x_{j+1} - x_j$ then for $\pi f$ a cubic spline interporlant over the mesh then for some constants $C_k$
$$ \| (f - \pi f)^{(r)} \|_\infty \leq C_r \|f^{(4)}\|_{\infty}h^{4-r} \quad 0 \leq r \leq 3$$
I would like to know if it is possible to obtain rates when $h \to 0$ for $$\| (f - P(f))^{(r)} \|_\infty$$
This could be easily obtained if a result such as the first one holds for higher derivatives i.e $$\|P(f)^{(r)}\|_\infty \leq C_{k,r}\|f^{(r)}\|.$$
Partial results are also useful. Could be for quadratic or cubic splines and for first or second derivative.
