Multivariate analogue of Jackson's inequality's modulus of continuity form

Question

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.

Answer 1

Cartwright and Kucharski 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

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

$$\lVert f - K_r * f \rVert_\infty \leq M_n \sup_{d(z,w) \leq \frac{1}{r}} \lvert f(z) - f(w) \rvert$$