A Survey of Results on the Limit q-Bernstein Operator

Abstract

The limit $q$-Bernstein operator $B_q$ 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 $B_q$ is a positive shape-preserving linear operator on $C[\mathrm{0,1}]$ with $\parallel B_q\parallel =1$. 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.