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I found a result of the estimation error of polynomial approximation in page 6 of https://scg.ece.ucsb.edu/publications/theses/ARajagopal_2019_Thesis.pdf

The statement is for $f \in W^{k, p}\left([-1,1]^q\right)$ and $n \geq 1$, there exists a constant $C=$ $C(k, p)$ and a polynomial $p_n$ of degree not exceeding $n$ such that $$ \left\|f-p_n\right\|_{L^p} \leq C\|f\|_{W^{k, p}} n^{-k} . $$ I think it is a simplifed version of multivariate Jackson's inequality, but I failed to find a reference of this result. Does anyone know the reference? Thanks.

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You can find this in 'Weakly Differentiable Functions' by Ziemer, Corollary 4.2.3, page 184. In this corollary, $k$ is the degree of the approximating polynomial and $0\le k < m$, ($m$ is what you have called $k$). Please note that the statement of the corollary contains a minor typo: the norm on the right-hand-side should be $||D^{k+1}u||_{m-(k+1),p;\Omega}$

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  • $\begingroup$ Thanks for your reply! Could you please explain more about how to transform the corollary to the statement above? In the corollary, k is the smoothness but here k is the order. Maybe I missed something. Thanks $\endgroup$
    – Iris
    Commented May 17 at 8:52
  • $\begingroup$ I really appreciate your help. But I still failed to derive it. (i) After taking $k=m-1$, the right hand side becomes $C(\frac{\|T\|}{T(1)})^m \|D^m u\|_{p,\Omega}$, what is the expression of $T$? And how to get the $n^{-m}$ term? (ii) Since $p^*>p$, we have $\|\cdot\|_{p^*}<\|\cdot\|_{p}$, then how to derive the left hand side? (iii) Does the result allow $p=\infty$? Sorry I am not quite familar with the functional analysis, could you please explain more? $\endgroup$
    – Iris
    Commented May 17 at 12:07
  • $\begingroup$ Auctually I am stuck in how to choose $T$. Also, I think if $p\leq r$, then $\|\cdot\|_p\geq \|\cdot\|_r$ $\endgroup$
    – Iris
    Commented May 17 at 14:01
  • $\begingroup$ $T$ is a functional you can choose, for example choose $T(f):=\int_\Omega f$, and $p=\infty$ is allowed. The correct inequality really is $\|\cdot\|_{p} \le K \|\cdot\|_{p^*}$ where the constant $K>0$ depends on the measure of the domain. $\endgroup$
    – ResearchMath
    Commented May 17 at 16:51

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