I found the multidimensional Jackson-type inequality for Lipschitz-continuous function approximation with algebraic polynomials. Here, $P_{n,m}$ is the set of algebraic polynomials of maximum degree $n$ and $m$ variables.
Let $c$ be a constant depending only on $f$, then $$f\in W^{k,\infty}(S) \Rightarrow \text{inf}_{p\in P_{n,m}} \left\lVert f-p \right\rVert_{L^2}\leq c n^{-k}.$$
I don't know if it is important but $S\subseteq\mathbb{R}^m$ is a compact set.
Could you help me proving such an equation? The author refers to Theorem 2 in [1] but I have trouble understanding it.
It seems the wikipedia article on Jackson's inequality leads to inequalities for $\text{inf}_{p\in P_{n,m}} \left\lVert f-p \right\rVert_{L^\infty}\leq C L_f n^{-k}$ where $C$ does not depend on $f$ or $n$ and $L_f$ depends on the Lipschitz constant of $f$ and its derivatives. However, I have only seen thin in the 1D case.
Would I now just use the continuous embedding of $L^\infty$ in $L^2$ leading to $$\left\lVert f\right\rVert_{L^2}\leq d \left\lVert f\right\rVert_{L^\infty}?$$ This seems a bit crude for me.
I have one more general questions to improve my intuition about such topics. - In most cases Jackson's Theorem is stated for periodic polynomials. Is it generally correct to assume that the inequalities are also valid for algebraic polynomials maybe on bounded or even compact domains?
[1] Ganzburg, Multidimensional Jackson theorems, 1981