Approximation of functions defined on the real axis by means of de la Vallée-Poussin operators
Let \(F\) be the class of functions \(\psi\) with \(\int^ \infty_ 1 (\psi(t)/t) dt< \infty\), where \(\psi(0)=0\), \(\psi\) is convex on \([1,\infty)\), vanishes at infinity, and \(\psi'(v)=\psi'(v+ 0)\) exists and is of bounded variation on \([0,\infty)\). If \(\beta\) is a fixed number then let \[ \widehat\psi(t)= \widehat\psi_ \beta(t)= {1\over\pi} \int^ \infty_ 0 \psi(v)\;\cos\left(vt+ {\beta\pi\over 2}\right) dv. \] \(\widehat C^ \psi_{\beta,\infty}\) denotes the class of functions of the form \(f(x)= A_ 0+ \int^ \infty_{-\infty} \varphi(x+ t)\widehat\psi(t) dt\) with \(A_ 0\) a given constant and \(\varphi\) functions on \(\mathbb{R}\) with \(\text{ess sup}|\varphi|\leq 1\). If \(\psi\in F\), then \(\widehat\psi\) is summable on \(\mathbb{R}\) and \(\widehat C^ \psi_{\beta,\infty}\) consists of continuous functions. In this paper asymptotic expressions are obtained for the upper bound of the deviation of de la Vallée Poussin operators from functions in the class \(\widehat C^ \psi_{\beta,\infty}\).
scientific article; zbMATH DE number 146530