We consider the Lorentz space Lq,τ( \({\mathbb{T}}^{m}\) ) of periodic functions of m variables and the class \({\text{W}}_{q,\tau ,\theta }^{a,b\left(\cdot \right),\overline{r} }\) with 1 < q, τ < ∞, a > 0, 1 < θ ⩽ ∞, where b(t) is a weakly oscillating function on [1, ∞). We find a sufficient condition on a function f ∈ \({L}_{q,{\tau }_{1}}\) ( \({\mathbb{T}}^{m}\) ) to belong to the space \({L}_{q,{\tau }_{2}}\) ( \({\mathbb{T}}^{m}\) ) with 1 < q < p < ∞, 1 < τ1 < τ2 < ∞ and the embedding \({\text{W}}_{q,{\tau }_{1},\theta }^{a,b\left(\cdot \right),\overline{r} }\) ⊂ \({L}_{q,{\tau }_{2}}\) ( \({\mathbb{T}}^{m}\) ). We obtain upper bounds for the best M-term trigonometric approximations of functions of class \({\text{W}}_{q,{\tau }_{1},\theta }^{a,b\left(\cdot \right),\overline{r} }\) in the \({L}_{q,{\tau }_{2}}\) ( \({\mathbb{T}}^{m}\) )-norm in the case 1 < q < 2 < p < ∞ under some conditions on a, τ1, τ2. For some values of a, τ1, τ2, the accuracy of the upper bound is established.
