A Survey of Results on the Limit -Bernstein Operator

Abstract

The limit -Bernstein operator emerges naturally as a modification of the Szász-Mirakyan operator related to the Euler distribution, which is used in the -boson theory to describe the energy distribution in a -analogue of the coherent state. At the same time, this operator bears a significant role in the approximation theory as an exemplary model for the study of the convergence of the -operators. Over the past years, the limit -Bernstein operator has been studied widely from different perspectives. It has been shown that is a positive shape-preserving linear operator on with . Its approximation properties, probabilistic interpretation, the behavior of iterates, and the impact on the smoothness of a function have already been examined. In this paper, we present a review of the results on the limit -Bernstein operator related to the approximation theory. A complete bibliography is supplied.

The distance between two limit $q$-Bernstein operators

“…See, for example, [8,12,13]. The review of obtained results along with extensive bibliography can be found in [10].…”

“…Here, (x − t) + is considered as a function of x. Plain calculation of R a,b ((x − t) + ) using (10) results in (12).…”

Approximating B-continuous functions using GBS operators of Bernstein–Schurer–Stancu type based on q-integers

“…A collection of results concerning the q-Bernstein polynomials thanks to a number of authors, is presented by Phillips, in [23]. In [20], Ostrovska studied the relation between the limit q-Bernstein operator and the theory of q-Bernstein polynomials. For the present study, we first remind some important elements of q-calculus.…”