Bounds for Fourier widths of classes of periodic functions with a mixed modulus of smoothness

Order-exact bounds are obtained for Fourier widths of the Nikol'skii-Besov classes $\mathrm{SB}_{p\theta}^{\Omega,l} (\mathbb{T}^d)$ and Triebel-Lizorkin classes $\mathrm{SF}_{p\theta}^{\Omega,l} (\mathbb{T}^d)$ of functions with a given majorant $\Omega$ for the mixed modulus of smoothness of order $l$ in the space $L_q(\mathbb{T}^d)$ for all relations between the parameters $p$, $q$, and $\theta$ under some conditions on $\Omega$. The upper bounds follow from order-exact bounds for approximations of the classes $\mathrm{SB}_{p\theta}^{\Omega,l} (\mathbb{T}^d)$ and $\mathrm{SF}_{p\theta}^{\Omega,l} (\mathbb{T}^d)$ by special partial sums of Fourier series in the multiple system $\Psi_d$ of periodized Meyer wavelets.