Best Constants in Preservation Inequalities Concerning the First Modulus and Lipschitz Classes for Bernstein-Type Operators

Abstract

We consider families (L t , t # T) of positive linear operators such that each L t is representable in terms of a stochastic process starting at the origin and having nondecreasing paths and integrable stationary increments. For these families, we give probabilistic characterizations of the best possible constants both in preservation inequalities concerning the first modulus and in preservation of Lipschitz classes of first order. As an application, we compute such constants for the Bernstein, Sza sz, Gamma, Baskakov, and Beta operators.1998 Academic Press

On Certain Best Constants for Bernstein-Type Operators

“…Conditions (H 1 ) and (H 3 ) correspond to conditions (B) and (C) in [3], and it should be observed that such conditions already imply the assumption (A) in the same paper.…”

“…The usual analytic definition of each operator will be accompanied by a specific probabilistic representation useful for our purposes. Such representations have been already used in other works; see, for instance, [1][2][3][5][6][7]. In view of what is said in Remark 3 above, we will only discuss the questions concerning W*.…”

“…To show the converse inequality, we adapt an argument taken from [3]. For each 0 < e < x, we define w e ( · ) in the following way…”

Best constants for tensor products of Bernstein type operators

“…Problems of this kind have been discussed in several works by using different approaches (see, for instance, [1][2][3][4][5][6][7][8][9][10]12] and references therein). The probabilistic approach developed in [1,2,[6][7][8][9]12] has proved to be suitable and fruitful when dealing with operators of probabilistic type (also called Bernstein-type operators), that is, operators allowing for a representation of the form…”

“…The probabilistic approach developed in [1,2,[6][7][8][9]12] has proved to be suitable and fruitful when dealing with operators of probabilistic type (also called Bernstein-type operators), that is, operators allowing for a representation of the form…”