The inverse theorem in various metrics of approximation theory for periodic functions with monotone Fourier coefficients

We prove the exactness with respect to order of an upper bound for the $k$th-order modulus of smoothness in $L_q({\mathbb T})$ in terms of the elements of a sequence of best approximations in $L_p({\mathbb T})$ on the class of all functions with monotonically decreasing Fourier coefficients, where $1<p<q<\infty$ and $k\in {\mathbb N}$.