Approximation properties of higher derivatives of periodic functions
If \(F_n\searrow 0\), let \(F_c\) denote the subclass of \(C_{2\pi}\) having the property \(E_n(f)\leq F_n\). The author proves that along \(f\in F_c\), \[ \sup E_n(f^{(r)})\sim n^rF_n+ \sum^\infty_{\nu=n+1} \nu^{r-1}F_\nu,\quad r= 1,2,\dots, \] if \(\sum\nu^{r- 1}F_\nu<\infty\). The same statement is proved for \(\widetilde f^{(r)}\), too. He also proves that \[ \sup\omega_k(f^{(r)}, n^{-1})\sim n^{-k} \sum^k_{\nu= 1}\nu^{k+ r-1} F_{\nu-1}+ \sum^\infty_{\nu= n+1} \nu^{r-1} F_\nu, \] and this relation holds for \(\widetilde f^{(r)}\) as well. Six further similar theorems are established for functions \(f\in H_\ell(\omega)\), i.e., if \(\omega_\ell(f,\delta)\leq \omega(\delta)\).
scientific article; zbMATH DE number 1256808