# Spline Approximation Results in $L^2(\mathbb{R})$ norm

I have seen in several papers that one can approximate a function $f \in C^{(k-1)}$ via splines in $S_\pi^k$ of order $k$ with extended knot sequence $\pi$ using a local approximation operator $Q: C^{(k-1)} \longrightarrow S_\pi^k$ which reproduce polynomials. However, I noticed that most of the results on the error bounds come in the $L^\infty$-norm, that is, $\|f - Qf\|_\infty = O(|\pi|^k).$ Are there any results on local spline approximation methods that come in the $L^2$-norm, and, possibly with similar behaviour?

Thank you very much!