Methods of trigonometric approximation and generalized smoothness. II

The paper deals with the equivalence of approximation errors in $L_p$-spaces ($0<p<\infty$) with respect to approximation processes, generalized $K$-functionals and appropriate moduli of smoothness. The results are used to derive various characterizations of periodic Besov spaces by means of constructive approximation and moduli of smoothness. The main focus lies on spaces $\mathbb{B}_{p,q}^s(\mathbb{T}^d)$, where $0 < p < 1$, $0 < q \leqslant\infty$ and $s > 0$.