Characterization of the saturation class \(C_ 0^{\psi}L_{\infty}\) (Q1101634)

Using some ideas of \textit{S. B. Stechkin} [Izv. Akad. Nauk SSSR, Ser. Mat. 15, 219-242 (1951; Zbl 0042.300)], the author improves a result of \textit{A. I. Stepanets} and \textit{A. K. Kushpel} [Preprint, Akad. Nauk Ukrain. SSR, Inst. Mat. (Kiev (1984)]. In fact he shows the following main theorem: If \(\psi\) is of class \(C^{(2)}\) on [1,\(\infty)\) and \(\psi\in {\mathcal M}_ C\), then \(f\in C_ 0^{\infty}L_{\infty}\) if and only if for every n, \(n=1,2,...\), there is a trigonometric polynomial \(T_ n\) *(f,x) of degree n-1 such that \(\| f-T_ n\quad *(f)\|_ C\leq A_ 1\psi (n),\| T^{\psi}_{n0}(f)\|_ C\leq A_ 2,\) \(n=1,2,..\). In addition, for every trigonometric polynomial \(T_ n\) holds the inequality \(\| T^{\psi}_{n0}\|_ C\leq A_ 3/\psi (n)\| T_ n\|_ C.\) Here \(A_ 1,A_ 2,A_ 3\) are constants nondepending on n, C is the space of continuous, \(2\pi\)-periodical functions, \({\mathcal M}_ C\) and \(C^{\psi}_{\beta}L_{\infty}\) are the spaces considered by \textit{A. I. Stepanets} in [Dokl. Akad. Nauk 277, 1074-1077 (1984; Zbl 0587.42001)] and A. I. Stepanets and A. K. Kushpel in the above mentioned work.