Abstract :
In this paper we determine the Lipschitz]Nikolski˘ı constants for the
Trotter]Feller operator which contains various well-known operators. As corollaries
to our general settings we obtain the Lipschitz]Nikolski˘ı constants for a series
of concrete operators including the Bernstein, Sz´asz, Gamma, Weierstrass, and
Baskakov operators, their generalizations such as the Cheney]Sharma and
Bleimann]Butzer]Hahn operators, and many others. Our results also improve
Rathore’s on the Meyer-K¨onig and Zeller operator and the Gamma operator of
M¨uller as well as Rathore and Singh’s on the Post]Widder operator. Throughout
the paper the probabilistic method is used intensively while a result in probability
theory on normal approximation plays a key role.
