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Let $f$ be a function defined on $[-1,1]^d$. Assume that all partial derivatives of $f$ up to order $r$ are continuous; and the $\infty$-norm of these partial derivatives are uniformly upper bounded by a constant. Let $p^*_n$ be the best degree $n$ approximation polynomial of $f$. That is,

$ p_n^* = \mathrm{argmin}_{p_n \in \mathcal{P}_n} \|f-p_n\|_{\infty}, $

where $\mathcal{P}_n$ is the set of all polynomials of degree n. I am interested in the rate of approximation.

For the single variable case, i.e., $d=1$, it is known that $\| f-p_n \|_{\infty} = \mathcal{O}(n^{-r})$. ($r$ is the order of smoothness defined above.)

My question is what is the rate of approximation when $d>1$.

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Short answer: The estimate is similar to that for functions in one variable.

Longer answer: The estimates of best approximation of a real-valued smooth function (by algebraic as well as by trigonometric polynomials) in terms of moduli of continuity of its derivatives are known as Jackson theorems. They were first proved for functions on the unit interval by Dunham Jackson. For functions on the $n$-cube a result of this type was obtained by D. J. Newman and H. S. Shapiro [On approximation theory (Oberwolfach, 1963), pp. 208–219, Birkhäuser, Basel, 1964; MR0182828 (32 #310)]. I cannot give now an exact quotation, because I do not remember the result well and do not have an access to the paper. There were subsequent generalizations, many by M. Gansburg. One of the newer results is the following theorem from MR1920286 Bagby, T.; Bos, L.; Levenberg, N.; Multivariate simultaneous approximation. Constr. Approx. 18 (2002), no. 4, 569–577:

If K is a connected, compact set in $\mathbb{R}^N$ such that every pair of distinct points $a,b$ of $K$ can be joined by a rectifiable arc in $K$ with length at most $s|a-b|$, where $s$ is a positive constant, and $f$ is a function of class $\mathcal{C}^r$ on an open neighborhood of $K$, then for each non-negative integer $n$, there exists a polynomial $p_n$ of degree at most $n$ defined on $\mathbb{R}^N$ such that for each multi-index $\alpha$ with $|\alpha|=\min\{r,n\}$, $$\sup_K |D^{\alpha}(f-p_n)|\leq \frac{C}{n^{r-|\alpha|}}\sum_{|\gamma|\leq r}\sup_K|D^{\gamma}f|,$$ where $C$ is a positive constant depending only on $N$, $r$ and $K$.

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