I would like to find a good reference for the following or a similar, probably well-known, approximation error result:

Let $\Omega\subset \mathbb{R}^d$ be bounded, $p\in [1,\infty]$, $l, m\in \mathbb{N}$ with $l\leq m$ and $(Q^h)_{h>0} \colon W^{m,p}(\Omega) \rightarrow W^{l,p}(\Omega)$ a family of operators such that for $u\in W^{m,p}(\Omega)$, $x\in \Omega$ and $h>0$ the value $Q^hu(x)$ only depends on $u$ in the ball $B_h(x)=\{y\in \Omega| |y-x|<h\}$. Furthermore we assume that $Q^h$ is exact for all polynomials of degree smaller than $m$ and that there exist a constants $K,C_1>0$ such that for all $x\in \Omega$ and $h>0$ we have $$ |Q^hu|_{W^{l,p}(B_h(x))}\leq C_1 |u|_{W^{l,p}(B_{Kh}(x))}. $$ Then there exist $C_2>0$ such that we have $$ \left|u-Q^hu\right|_{W^{l,p}}\leq C_2h^{m-l}|u|_{W^{m,p}} \text{ for all } u\in W^{m,p} \text{ and } h>0. $$ The estimate can be proven using the Bramble-Hilbert lemma.

In finite element books one can find similar estimates but not in this full generality. Jackson's inequality is also similar to this but there it is with trigonometric polynomials. If it helps in my case one has to additionally assume $m>d/p$ because my operator is defined only for continuous functions.


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