On Certain Best Constants for Bernstein-Type Operators

Abstract

We discuss the generalized version of a best-constant problem raised by Z. Li in a note which recently appeared in Journal of Approximation Theory. Some best constants for well known Bernstein-type operators are obtained.

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…”

“…We actually have, by the strong law of large numbers for Poisson processes, where {N(u): u 0} is a standard Poisson process, {γ t : t 0}, {γ t : t 0}, and {γ t : t 0} are standard gamma processes, and these four processes are supposed to be mutually independent and defined on the same probability space. It should be observed that P t,r is the composition B * t • H r of the beta-type operator B * t given in (11) with the Baskakov operator H r given in (12). When r is a positive integer, P t,r becomes a modified version of an operator introduced by Stancu [16].…”

Global smoothness preservation with second order modulus of smoothness

“…Best constants for Bernstein-type oparators with the least concave majorant are obtained in de la Cal and Cárcamo [7]. In the multivariate setting, best constants in preservation inequalities for Bernstein-type operators can be found in de la Cal and Valle [10,11], de la Cal et al [9] and de la Cal and Cárcamo [8].…”

Approximation Properties of Parametric Kantorovich-Type Operators on Half-Bounded Intervals

“…At the same time, operator plays important role in computer-aided geometric design, which deals with computational aspects of geometric objects in mathematical developments of curves and surfaces. Inspired by the ideas of Zhang Chungou and others [16][17][18][19][20][21][22][23][24], some shape-preserving properties of these operators are obtained.…”