Jackson-type inequalities and widths of function classes in \(L_{2}\)
The paper is devoted to the Jackson-type inequalities \[ E_{n-1}(f) \leq Cn^{-r}\omega_k (f^{(r)}, t/n), \qquad t>0, \] where \(E_{n-1}(f)\) is the error of the best approximation of \(f\) in \(L_2[0, 2\pi]\) by trigonometric polynomials of order \(\leq n-1\) and \(\omega_k\) stands for the \(k\)th modulus of continuity. For \(0\leq t \leq \pi/n\), upper and lower estimates are obtained for the quantity \[ \sup_f {E_{n-1}(f) \over \omega_k (f^{(r)}, t)}, \] where the supremum is taken over all non-constant \(f\) having an absolutely continuous \((r-1)\)th derivative with \(f^{(r)}\in L_2\). A similar supremum, \[ \sup_f {n^{2r}E_{n-1}^2(f) \over \left ( \int_0^t \omega_k^{2/k} (f^{(r)},x)\,dx \right )^k}, \qquad n,k \in {\mathbb N}, \quad 0<t\leq \pi/(2n), \] is computed exactly. In the second part of the paper, \(n\)-widths are evaluated of two classes of functions \(f\) defined by restrictions on \(\omega_k(f)\). In the case \(k=1\) the results of the paper were obtained earlier by Taikov.
scientific article; zbMATH DE number 5155490