= Multivariable Jackson approximation
{title2=$\|f-P_N\|_\infty\le C_{n,p}N^{-p}\|f\|_{C^p}$}
For $p\ge1$ and $f\in C^p(\mathbb T^n)$, there is a real <trigonometric polynomial> with coordinate frequencies $|k_j|\le N$ whose uniform error is at most $C_{n,p}N^{-p}\|f\|_{C^p}$. One obtains the multivariable statement by applying bounded one-dimensional Jackson approximation operators successively in each coordinate and telescoping their errors. Each operator commutes with differentiation in the other coordinates, and its uniform operator norm is bounded independently of $N$; the total error is bounded by a constant times $N^{-p}\sum_j\|\partial_j^p f\|_\infty$.
Back to article page
