Rational interpolation Fejér and de la Vallée-Poussin type operators (Q1326038)

Given a function \(f\) on \(\mathbb{R}\) and a sequence of complex numbers \(\{z_ k\}^ \infty_{k=1}\) in the upper half-plane, rational interpolation operators of Fejér and de la Vallée-Poussin type are constructed, proceeding on the basis of the Bernstein sine and cosine quotients for \(\{z_ k\}\). For \(f\in C_ \infty\) (i.e. \(f\in C(-\infty, +\infty)\) and there exist finite \(\lim_{x\to -\infty} f(x)=\lim_{x\to +\infty} f(x))\) an estimate for the rate of the uniform convergence on \(\mathbb{R}\) of de la Vallée-Poussin type rational functions to \(f\) is given. Moreover, under additional assumption on \(\{z_ k\}\) the uniform convergence on \(\mathbb{R}\) of Fejér type rational functions to \(f\in C_ \infty\) is also established.