Approximation properties of modified gamma operators (Q884331)

Let \( f \) be a real function on \( (0, \infty) \) satisfying the growth conditions \[ \begin{align*}{ \vert f(t) \vert & \leq M_1e^{\beta /t} , \quad \beta \geq 0,\; M_1 > 0,\; t \to 0, \cr \vert f(t) \vert & \leq M_2 t^p , \quad M_2 > 0,\; p \in I\!\!N ,\; t \to +\infty. \cr}\end{align*} \] The modified Gamma operators \( G_2 \) applied to \( f \) are given by \[ G_n (f; x) = \int^\infty_0 \varphi_n (x; u) f \left({n\over u} \right) du, \qquad n \geq p, \] where \( \varphi_n (x; u) := e^{-ux} \; u^n {x^{n+1}\over n!}.\) In this paper the pointwise approximation of modified Gamma operators \(G_n\) is studied for the function classes \( \Phi_B \) and \( \phi_{DB},\) given by \[ \begin{align*}{ \Phi_B & := \{f \mid f \text{ is bounded on every finite subinterval of }(0, \infty)\}, \cr \Phi_{DB} & := \{f \mid f(x) = \int^x_1 h(t) \,dt + f(1) ; 0 < x < \infty,\ h \in \Phi_B \}. \cr}\end{align*} \]