Results on asymptotic characteristics of classes of functions with mixed smoothness are obtained in the paper. Our main interest is in estimating the Kolmogorov widths in the uniform norm of classes with small mixed smoothness. We prove the corresponding bounds for the unit balls of the trigonometric polynomials with frequencies from a hyperbolic cross. We demonstrate how our results on the Kolmogorov widths imply new upper bounds for the optimal sampling recovery in the $L_2$ norm of functions with small mixed smoothness.

中文翻译：

---

Kolmogorov 宽度的界限和混合平滑度小的类的采样恢复

文中得到了具有混合平滑度的函数类的渐近特性的结果。我们的主要兴趣是在具有小的混合平滑度的类的统一范数中估计 Kolmogorov 宽度。我们证明了三角多项式的单位球的相应边界，其频率来自双曲线交叉。我们展示了我们在 Kolmogorov 宽度上的结果如何暗示最佳采样恢复的新上限${升}_2$ 具有小的混合平滑度的函数范数。