Studying Baskakov--Durrmeyer operators and quasi-interpolants via special functions

We prove that the kernels of the Baskakov-Durrmeyer and the Szasz-Mirakjan-Durrmeyer operators are completely monotonic functions. We establish a Bernstein type inequality for these operators and apply the results to the quasi-interpolants recently introduced by Abel. For the Baskakov-Durrmeyer quasi-interpolants, we give a representation as linear combinations of the original Baskakov-Durrmeyer operators and prove an estimate of Jackson-Favard type and a direct theorem in terms of an appropriate K-functional.