A Survey of Results on the Limit q-Bernstein Operator

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 exemplarymodel 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[0, 1] with ‖B q ‖ = 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.