Asymptotic behavior of the Gelfand and the Bernstein widths of some classes of analytic functions
Let \(H^p\), \(1\leq p\leq\infty\), be the Hardy class in the unit disk \(\mathbb{D}\), and let \(T_r\) be the circle \(\{z:|z|=r\}\), \(0<r\leq 1\), \(\mathbb{T}= T_1\). The author studied the Gelfand widths of the unit ball \(BH^p\) in the metric of the space \(L_q(T_r,\mu)\) for \(p\geq q\) and for an arbitrary measure \(\mu\) on \(T_r\). Fisher and Stessin proved that for \(p\geq q\) the Kolmogorov, the Gelfand, and the linear widths of \(BH^p\) in the space \(L_q(K,\mu)\) (when \(K\) is a compact set in \(\mathbb{D}\)) coincide. For \(p<q\), very little is known about the widths in question. The author established Ismagilov type theorems for the Gelfand and the Bernstein widths. From the results of [the author, Mat. Sb., n. Ser. 117(159), No. 2, 279-285 (1982; Zbl 0488.30027) and Zap. Nauchn. Sem. 217, 112-129 (1994)] one can obtain nontrivial lower bounds for the Gelfand widths and upper bounds for the Bernstein widths. However, it is difficult to derive the reverse estimates from these results. The present paper deals with just this problem.
scientific article; zbMATH DE number 1002988