According to Jackson's 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.