On Jackson inequality in $L_p(\mathbb T^d)$

The author proved some necessary and sufficient conditions on a finite set of $d$–dimensional vectors $\{\alpha_l\}$, when Jackson–Youdin inequality for the approximation of periodic function $f$ by trigonometric polynomials:
$$ E_{n-1}(f)_q\le A\cdot n^{-r +(d/p-d/q)_+}\cdot \max\limits_{l}\|\Delta_{2\pi\alpha_l/n}^m f^{(r)}\|_p, $$
where $A>0$ is independent of $f$ and $n$, holds. A criterion of solvability of the homological equation
$$ f(x)-\frac{1}{(2\pi)^d}\int f(t)dt=\varphi(x+2\pi\alpha)-\varphi(x)\qquada.e.\ x $$
on the sets of functions $\{f\colon\ f^{(r)}\in L_p(\mathbb T^d)\}$ is obtained.