Jesús de la Cal

A note on limiting properties of some Bernstein-type operators

Abstract Some well-known Bernstein-type operators are exhibited as limits, in an appropriate sense, of other ones. This is readily made by using limit theorems of probability theory. Moreover, in two cases, rates of convergence are also obtained by using probabilistic tools.

On uniform approximation by some classical Bernstein-type operators☆

Abstract We investigate the functions for which certain classical families of operators of probabilistic type over noncompact intervals provide uniform approximation on the whole interval. The discussed examples include the Szasz operators, the Szasz–Durrmeyer operators, the gamma operators, the Baskakov operators, and the Meyer–Konig and Zeller operators. We show that some results of Totik remain valid for unbounded functions, at the same time that we give simple rates of convergence in terms of the usual modulus of continuity. We also show by a counterexample that the result for Meyer–Konig and Zeller operators does not extend to Cheney and Sharma operators.

Preservation of Moduli of Continuity for Bernstein-Type Operators

It is well known that many Bernstein-type operators preserve some properties of the functions on which they act, such as monotonicity, convexity, Lipschitz constants, etc. (cf. for instance [2]). In this paper, attention is focused on preservation of global smoothness, as measured by the usual moduli of continuity of first and second order. To the best of our knowledge, this problem has been studied by Kratz and Standtmuller in [11] for the first time. In this work the authors consider sequences (L n ) n≥1 of one-dimensional descrete operations satisfying certain moment assumptions and obtain estimates of the form

On the uniform convergence of normalized Poisson mixtures to their mixing distribution

\omega \left( {L_n f;h} \right) \leqslant c\omega \left( {f;h} \right),

BEST CONSTANTS FOR TENSOR PRODUCTS OF BERNSTEIN, SZASZ AND BASKAKOV OPERATORS

(1) where ω(f;.) stands for the usual first modulus of continuity of function f and c is a positive constant which depends on the particular family of operations considered, but not upon f nor n and h. They provide the estimate c ≤ 4 in some important examples, such as Bernstein, Szasz and Baskakov operators.

A best constant for bivariate Bernstein and Szász-Mirakyan operators

Let F be the gamma distribution function with parameters a > 0 and α > 0 and let G s be the negative binomial distribution function with parameters α and a / s, s > 0. By combining both probabilistic and approximation-theoretic methods, we obtain sharp upper and lower bounds for . In particular, we show that the exact order of uniform convergence is s –p , where p = min(1, α ). Various kinds of applications concerning charged multiplicity distributions, the Yule birth process and Bernstein-type operators are also given.

On Certain Best Constants for Bernstein-Type Operators

Under fairly general assumptions, we obtain the exact order of convergence, in the uniform distance, of normalized Poisson mixtures to their mixing distribution. This entails giving sharp rates of convergence in the classical inversion theorem for Laplace transforms.

Limiting properties of Inverse Beta and generalized Bleimann-Butzer-Hahn operators

We establish a strong version of a known extremal property of Bernstein operators, as well as several characterizations of a related specific class of positive polynomial operators.