A note on rational and spline approximation (Q2640077)

In a recent technically involved but important paper (in my opinion), \textit{P. Petrushev} [(*) ibid. 50, 141-159 (1987; Zbl 0647.41018)] established a weak-type estimate of the rational approximation by the best spline approximation. This was done following the classical result of \textit{D. J. Newman} [Mich. Math. J. 11, 11-14 (1964; Zbl 0138.044)] about approximation of the function \(| x|\) by rational functions. In the present note, we make some observations that yield a substantial improvement in the main result of (*).