A Survey of Results on the Limit q-Bernstein Operator

Abstract EN

The limit q-Bernstein operator Bq emerges naturally as a modification of the Szász-Mirakyan operator related to the Euler distribution, which is used in the q-boson theory to describe the energy distribution in a q-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 q-operators.

Over the past years, the limit q-Bernstein operator has been studied widely from different perspectives.

It has been shown that Bq is a positive shape-preserving linear operator on C[0,1] with ∥Bq∥=1.

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 q-Bernstein operator related to the approximation theory.

A complete bibliography is supplied.