Estimates for the widths of classes of periodic functions of several variables – I

We establish estimates sharp in order for the Kolmogorov and linear widths of the classes $\mathrm{B}^{s\,m}_{p\,q}(\,\mathbb{T}^k)$ and $\mathrm{L}^{s\,m}_{p\,q}(\,\mathbb{T}^k)$ of Nikol'skii–Besov, Lizorkin–Triebel types respectively, in the space $L_r(\mathbb{T}^k)$ for a certain range of the parameters $s,p,q,r$, and $m$.