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Let $f$ be a function on $[-1,1]^d$ with some smoothness property, for example, it is in the Sobolev space $W^{k,p}$. Let $P_n$ is a space of polynomials with degree $n$. My question is what is the rate of the approximation error of $p_n\in P_n$. Here is a related question: Multivariate polynomial approximation of smooth functions

According to the results in A.F. Timan, Theory of Approximation of Functions of a Real Variable, there exists a polynomial $p_n$ such that $$ \left\|f-p_n\right\|_{L^p} \leq C\|f\|_{W^{k, p}} n^{-k} . $$ However, here the degree of the polynomial $n$ means the degree in each variable, not the total degree. I wonder if there exists a similar result where $n$ means the total order of the polynomial.

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  • $\begingroup$ Isn't total degree at most $d$ times the degree in each variable? So you will get an extra factor $d^{k}$, which can be absorbed by the constant $C$. $\endgroup$
    – Aleksei Kulikov
    Commented May 19 at 10:00
  • $\begingroup$ Can I understand as the follows: there exists a polynomial with the degree $\floor{\frac{N}{d}}$ in each variable, which has the total degree not exceeding $N$. Then we can apply the above result and obtain the inequlity with $n$ replaced by $N$. $\endgroup$
    – Iris
    Commented May 19 at 12:12
  • $\begingroup$ Yes, something like this. $\endgroup$
    – Aleksei Kulikov
    Commented May 19 at 12:16

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