Estimates of the Fourier widths of the classes of periodic functions with given majorant of the mixed modulus of smoothness

We obtain some order-sharp estimates for the Fourier widths of Nikol'skii–Besov and Lizorkin–Triebel function classes with given majorant of the mixed modulus of smoothness in the Lebesgue space for a few relations between the parameters of the class and the space. The upper bounds follow from estimates of the approximation of functions of these classes by special partial sums of their Fourier series with respect to the multiple system of periodized Meyer wavelets.