Inequalities for the best “angular” approximation and the smoothness modulus of a function in the Lorentz space

In this paper, we consider the Lorentz space $L_{p, \tau}(\mathbb{T}^{m})$ of $2\pi$-periodic functions of several variables, the best “angular” approximation of such functions by trigonometric polynomials, and the mixed smoothness modulus of functions from this space. The properties of the mixed smoothness modulus are given and strengthened versions of the direct and inverse theorems on the “angular” approximations are proved.