Approximation properties of the Rogosinski integrals defined on the real line (Q1370578)

Let \(f\) belong to the class \(T^1\), which means that \(f\) is a (complex valued) measurable function on the real line such that \(|f(t) |(1+t^2)^{-1}\) is integrable on \(\mathbb{R}\). The Rogosinski integrals \(R_\lambda f\), defined by \[ (R_\lambda f)(z): =\lambda \int^\infty_{-\infty} f(u) \psi \bigl(\lambda (z-u) \bigr)du \] for \(\lambda >0\) and \(z\in \mathbb{C}\), where \(\psi (\zeta): ={1\over 2} {\cos (\zeta) \over 2(\pi/2)^2 -\zeta^2}\), are entire functions of exponential type \(\leq\lambda\). The author proves several results concerning the approximation of \(f\) by \(R_\lambda f\) for functions \(f\) belonging to certain subclasses of \(T^1\). Moreover, asymptotic formulas are obtained for the ``worst case''-order of approximation for some subclasses of \(T^1\) and for \(\lambda\) tending to infinity.