Entropy Numbers and Widths for the Nikol’skii–Besov Classes of Functions of Many Variables in the Space L∞

Abstract

We establish order estimates for the entropy numbers of the Nikol’skii–Besov classes $$B_{p,\theta}^r$$ of periodic functions of many variables in the metric of the space L∞. In the two-dimensional case the results obtained are used for establishing exact-order estimates of the Kolmogorov widths of the classes $$B_{p,\theta}^r$$ , 2 ≤ p ≤ ∞, and of the linear and the trigonometric widths of the classes $$B_{\infty,\theta}^r$$ in the metric of the space L∞.