On the simultaneous approximation of functions and their derivatives by the Szász-Mirakyan operator
The Szász-Mirakyan operator \(S_ n\) is defined by \(S_ n(f,x):=\sum^{\infty}_{k=0}f(k/n)p_ k(nx)\), \(p_ k(t)=e^{-t}t^ k/k!\). The class \(B_ r^{(\alpha)}:=\{f|\) \(f^{(r-1)}\in C[0,\infty)\), \(f_{\pm}^{(r)}(x)\) exist everywhere and are bounded on every finite subinterval of [0,\(\infty)\) and \(f_{\pm}^{(r)}(t)=O(t^{\alpha t})\) (t\(\to \infty)\) for some \(\alpha >0\}\), \(r=0,1,...\); \(f_{\pm}^{(0)}(x)\) means \(f(x\pm))\) is larger than the class of functions of generalized bounded variation. The author gives a precise estimate of the rate of convergence of the operator \(S_ n(f,x)\) to \((f_+^{(r)}(x)+f_-^{(r)}(x))\) for \(f\in B_ r^{(\alpha)}\).
functions of generalized bounded variation
scientific article; zbMATH DE number 4142715