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!


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