Korovkin type theorems for operators acting on functions of polynomial and exponential growth on $[0,\infty)$

We prove two Korovkin-type approximation theorems for sequences of positive linear operators acting on continuous functions on $[0,\infty)$. Under the assumption of pointwise convergence on suitable test functions, we establish pointwise convergence for all functions with polynomial or exponential growth. As direct applications, we obtain convergence results for the classical Baskakov and Sz\'asz--Mirakjan operators. The proposed method offers an elementary framework that can be applied to a broad class of positive linear operators.