Approximation of differentiable functions by functions of large smoothness

The order of the quantity $\delta(L)=\sup\limits_{x_1}\inf\limits_{x_2}\|x_1-x_2\|_{L_s[0,2\pi]}$ as $L\to\infty$ is studied for the classes of periodic functionsx $x_1\in\widetilde W_p^n(1)$, $x_1\in\widetilde W_q^n(L)$. Necessary and sufficient conditions under which the inequality
$$ \|x^{(n)}\|_{L_p}\le C\|x\|_{L_q}^\alpha\|x^{(m)}\|_{L_s}^\beta $$
with the constant independent of $x$ holds for all periodic functions x(t) with $\int_0^{2\pi}x(t)\,dt=0$ and $x^{(m)}(t)\in L_s[0,2\pi]$ are found.