On the best linear approximation methods and the widths of certain classes of analytic functions (Q1966187)

Let \( f_a^{(r)} \) be the \(r\)th-order derivative of the function \( f \) with respect to the argument \( x \) of the complex variable \( z=\rho e^{ix}. \) If \( f_a^{(r)}\) is in the Hardy space \( H_q, q\geq 1, \) then its second modulus of smoothness is denoted by \( \omega_2(f_a^{(r)},\cdot)_{H_q}. \) For an arbitrary increasing continuous function \( \Phi:\mathbb R_+\rightarrow \mathbb R_+\), \(\Phi(0)=0, \) and for the parameter \( \mu\geq 1/2, \) denote by \( W_a^rH_q(\Phi,\mu)\), \(r\in \mathbb N \) the class of functions \( f\in H_q \) for which \[ \frac 1t\int_0^t\omega_2(f_a^{(r)}, 2x)_{H_q} \biggl(1+(\mu^2 -1)\sin\frac{\pi x}{2t}\biggr) dx\leq \Phi(t), \quad t\in (0,\frac{\pi}2]. \] Let \( d^n(\mathcal M, X),\delta_n(\mathcal M,X)\) be respectively the Gelfand and linear \(n\)-widths of the convex, centrally symmetric compact set \( \mathcal M \) in the Banach space \( X. \) The author proves that under some conditions on \( \Phi \), when \( \mathcal M=W_a^rH_q(\Phi,\mu), \) and \( X=H_q \) we have \[ d^n(\mathcal M,X)=\delta_n(\mathcal M,X)=\frac{\pi}{2(\pi-2)n^r} \Phi\left(\frac{\pi}{2n\pi}\right). \] Exact values of the Kolmogorov and Bernstein \(n\)-widths are also obtained for other classes of analytic functions.