On the convergence and saturation problem of a sequence of discrete linear operators of exponential type in \(L_ p(-\infty,\infty)\) spaces
The author considers the following discrete linear operators of entire exponential type: \[ I_{\sigma}(f,x)=\sum^{\infty}_{k=- \infty}f(t_ k)\quad \sin^ 2[\frac{\sigma}{2}(x-t_ k)]/[\frac{\sigma}{2}(x-t_ k)]^ 2 \] where k is an integer, \(\sigma t_ k=2k\pi\) and \(f\in L_ p(-\infty,\infty)\). Operators of this type were considered by \textit{P. Butzer} [Constructive function theory, Proc. int. Conf., Varna/Bulg. 1981, 258-274 (1983; Zbl 0542.94004)]. Different types of estimates for \[ \| I_{\sigma}(f)-f\|_ p\quad (1\leq p<\infty) \] are given and the saturation problem for this approximation problem is solved.
discrete linear operators of entire exponential type
scientific article; zbMATH DE number 3994197
scientific article