Kolmogorov Widths of the Besov Classes $$B^1_{1,\theta}$$ and Products of Octahedra

Abstract

We find the decay orders of the Kolmogorov widths of some Besov classes related to $$W^1_1$$ (the behavior of the widths for the class $$W^1_1$$ remains unknown): $$d_n(B^1_{1,\theta}[0,1],L_q[0,1])\asymp n^{-1/2}\log^{\max\{1/2,1-1/\theta\}}n$$ for $$2<q<\infty$$ and $$1\le\theta\le\infty$$ . The proof relies on the lower bound for the width of a product of octahedra in a special norm (maximum of two weighted $$\ell_{q_i}$$ norms). This bound generalizes B. S. Kashin’s theorem on the widths of octahedra in $$\ell_q$$ .