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 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 [ 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. © 2013 sofiya ostrovska.