According to [Jackson's inequality](https://en.wikipedia.org/wiki/Jackson%27s_inequality), there is $c > 0$ s.t. for any continuous function $f: S^1 \rightarrow \mathbb{R}$ (where $S^1 := \mathbb{R} / \mathbb{Z}$ is the circle) and integer $n$ there is a trigonometric polynomial $q: S^1 \rightarrow \mathbb{R}$ of degree $n$ s.t.


$$\lVert f - q \rVert\_\infty \leq c \, \omega(f,\frac{1}{n})$$


Here $\omega$ is the modulus of continuity:


$$\omega(f,\delta):=\sup\_{\substack{x \in S^1 \\ h \in [0,\delta]}} \lvert f(x) - f(x + h) \rvert$$


I'm looking for a generalization of this theorem to functions $f: T^d \rightarrow \mathbb{R}$ where $T^d := \mathbb{R}^d / \mathbb{Z}^d$ is the canonical $d$-dimensional torus.



Multivariate analogue of Jackson's inequality's modulus of continuity form
[Cartwright and Kucharski](http://www.sciencedirect.com/science/article/pii/0021904588901025) give a generalization of Jackson's inequality for an arbitrary compact connected Lie group. I only need the uniform norm, rank 1 case for the canonical torus, so I state here this special case.


Theorem
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Fix $n \in \mathbb{N}$. Then there is $M\_n >0$ s.t. for any $r > 0$ there is $K\_r \in C^\infty(T^n, \mathbb{R})$ s.t. its Fourier transform satisfies $\operatorname{supp} \hat{K}\_r \subseteq \{\vartheta \in \mathbb{Z}^n \mid \lVert \vartheta \rVert \leq r\}$ and for any $f \in C(T^n, \mathbb{C})$ we have


$$\lVert f - K\_r \* f \rVert\_\infty \leq M\_n \sup\_{d(z,w) \leq \frac{1}{r}} \lvert f(z) - f(w) \rvert$$


