Lipschitz–Nikolskii Constants for the Trotter–Feller Operator

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In this paper we determine the Lipschitz–Nikolskii constants for the Trotter–Feller operator which contains various well-known operators. As corollaries to our general settings we obtain the Lipschitz–Nikolskii constants for a series of concrete operators including the Bernstein, Szász, 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-König and Zeller operator and the Gamma operator of Müller 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.