Best constants in preservation inequalities concerning the first modulus and Lipschitz classes for Bernstein-type operators
Many families \(\{L_t:t\in T\}\) of positive linear operators considered in the literature on approximation theory admit a probabilistic representation of the form \[ L_tf(x)= Ef\bigl(Z_t(x)\bigr) \quad (x\in I,\;t\in T),\tag \] where \(I=[0,1]\) or \(I=[0,\infty)\), \(T=\mathbb{N}\) or \(T=(0,\infty)\), \(E\) denotes mathematical expectation, \(\{Z_t(x):x\in I,\;t\in T\}\) is a doubly-indexed stochastic process of integrable random variables taking values in \(I\), and \(f\) is any real-valued measurable function on \(I\) for which the right-hand side of (*) is well defined. For such families the authors give probabilistic characterisations of the best possible constants both in preservation inequalities concerning the first modulus and in preservation of Lipschitz classes of first order. Such constants for the Bernstein, Szász, gamma, Baskakov and beta operators are computed. This continues a study begun in \textit{J. A. Adell} and \textit{J. de la Cal}, in Approximation, probability and related fields Plenum, New York, 1994, 1-18 (1994; Zbl 0863.41007)].
scientific article; zbMATH DE number 1197075