Approximation by Fourier sums and Kolmogorov widths for classes $\mathbf{MB}^\Omega_{p,\theta}$ of periodic functions of several variables

We obtain exact order estimates for approximations of mixed smoothness classes $\mathbf{MB}^\Omega_{p,\theta}$ by Fourier sums in the metric $L_q$ for $1<p<q<\infty$. The spectrum of approximation polynomials lies in the sets generated by level surfaces of the function $\Omega(t)/\prod_{j=1}^dt_j^{1/p-1/q}$. Under some matching conditions on the parameters $p,q$ and $\theta$, we obtain exact order estimates for Kolmogorov widths of the classes under consideration in the metric $L_q$.