On strong asymptotics of mean \(N\)-widths for classes of functions analytical on the real axis (Q677736)

Let \(1\leq p \leq \infty\) and \({\mathcal A}_p(D(h))\) denote the class of analytic in the strip \(D(h)=\{z=x+iy\in{\mathbf C}: |y|< h\}\) functions \(f\) which satisfy with any \(|y|<h\) the condition \(\left( \int_R |f(x+iy)|^p dx\right)^{1/p} \leq 1\). The restriction of the class \({\mathcal A}_p(D(h))\) on \({\mathbf R}\) is denoted by \({\mathcal A}_p^R (D(h))\). The author proves the following theorem: If \(\bar \Pi_n\) is either the mean Kolmogorov \(n\)-width or the mean Bernstein \(n\)-width then \[ \limsup_{n\to \infty} \left( \bar \Pi_n ({\mathcal A}_p^R(D(h)), L_p({\mathbf R})\right)^{1/n} = e^{-\pi h}. \]