Let $s(x)$ be the natural cubic spline interpolant of a function $f\in C^4$. There are known bounds on the $L^{\infty}$ error, $\|f^{(r)}(x) - s^{(r)} (x) \|_{\infty} $ for $r=0,1,2,3$. See Hall & Meyer, 1976, J. Approx. Theory.

My question: Are there any known optimal bounds on $\|f^{(r)}(x) - s^{(r)} (x) \|_{2} $? Under what conditions? Could you refer me?

If there are none, optimal rates could also help considerably.