Simultaneous approximation for the Phillips operators (Q885590)

Let denote the class \(B_{r,\alpha}\) by \(B_{r,\alpha}= \{f: f^{(n-1)}\in C[0,\infty)\), \(f^{(r)}_{\pm}(x)\) exist everywhere and are bounded on every finite subinterval of \([0,\infty)\) and \(f^{(r)}_{\pm}(x)= 0(r^{\alpha x})\) \((x\to\infty)\), for some \(\alpha> 0\}\), \(r\in\mathbb{N}\). By \(f^{(0)}_{\pm}(x)\) is denoted \(f(x\pm)\). for the Phillips operators \[ P_n(f, x)= n\sum^\infty_{k=1} p_{n,k}(x) \int^\infty_0 p_{n,k-1}(t) f(t)\,dt+ e^{-nx} f(0),\;n\in\mathbb{N},\;x\in [0,\infty), \] where \(p_{n,k}(x)= e^{-nx}{(nx)^k\over k!}\) and \(f\in B_{r,\alpha}\), are studied the simultaneous approximation properties by means of the decomposition technique for functions of bounded variation.