Multivariable Jackson approximation ($\Vert f-P_N{\Vert }_{\infty }\le C_{n,p}{N}^{-p}\Vert f{\Vert }_{C^p}$)

For $p\ge 1$ 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 $C_{n,p}{N}^{-p}\Vert f{\Vert }_{C^p}$. 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 $N^{-p}{\sum }_j\Vert {\partial }_j^{p}f{\Vert }_{\infty }$.