Let $\Omega \subset \mathbb{R}^d$ be compact and convex, and let $f \in C^{k+1}(\Omega)$. Let $P_k$ be the set of multivariate polynomials up to degree $k$ on $\Omega$. I am looking for any results in the literature that allow us to estimate $$ \inf_{p \in P_k} \|f - p\|_{L^2(\Omega)}$$ in terms of norms on the $(k+1)$th derivative. A classical estimate of this form occurs on the unit interval in 1D, where we have an explicit form for the orthogonal (Legendre) polynomials, but we don't have access to such formulas for an arbitrary domain $\Omega$. I suspect the result is out there, but I can only seem to find results related to $L^{\infty}$ approximation. Does anybody know where is the right place to look for such results?
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Check out Bramble-Hilbert Lemma: http://en.wikipedia.org/wiki/Bramble%E2%80%93Hilbert_lemma
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$\begingroup$ Great, thank you! From that article, I was able to jump-start my literature search to find sharper results, explicit constants, etc. $\endgroup$ Commented Sep 10, 2014 at 2:13