Approximation of the semi-infinite interval

The approximation of a function f ∈ C [ a , b ] by Bernstein polynomials is well-known. It is based on the binomial distribution. O. Szasz has shown that there are analogous approximations on the interval [ 0 , ∞ ) based on the Poisson distribution. Recently R. Mohapatra has generalized Szasz' result to the case in which the approximating function is α e − u x ∑ k = N ∞ ( u x ) k α + β − 1 Γ ( k α + β ) f ( k α u ) The present note shows that these results are special cases of a Tauberian theorem for certain infinite series having positive coefficients.