On the uniform approximation by generalized Bernstein-means (Q1187316)

Let \(S_ n[g]\) denote the trigonometric polynomials of degree at most \(n\) interpolating the function \(g\) in \(C_{2\pi}\) at \(m=2n+1\) equidistant nodes given by \[ t_ i^{(m)}=\tau+{2\pi i\over m},\;S_ n[g](t_ i^{(m)})=g(t_ i^{(m)})\quad (i\in\mathbb{Z}). \] The purpose of this note is to present some results on uniform approximation by the generalized Bernstein means defined by \[ B_{kn}[g](t)=2^{-k}\sum^ k_{j=0}S_ n[g]\left(t+{k-2j\over m}\pi\right)\quad (k=0,1,2,\ldots). \] For earlier work on this theme see \textit{O. Kis} and \textit{G. P. Névai} [Acta Math. Acad. Sci. Hungar 26, 385-403 (1975; Zbl 0332.42005)].