The Unicity Theorems for the Limit Q-Bernstein Operator

Abstract

The limit q-Bernstein operator [image omitted] emerges naturally as a q-version of the Szasz-Mirakyan operator related to the Euler distribution. The latter is used in the q-boson theory to describe the energy distribution in a q-analogue of the coherent state. The limit q-Bernstein operator has been widely studied lately. It has been shown that [image omitted] is a positive shape-preserving linear operator on [image omitted] with [image omitted] Its approximation properties, probabilistic interpretation, the behaviour of iterates, eigenstructure and the impact on the smoothness of a function have been examined. In this article, we prove the following unicity theorem for operator: if f is analytic on [0, 1] and [image omitted] for [image omitted] then f is a linear function. The result is sharp in the following sense: for any proper closed subset [image omitted] of [0, 1] satisfying [image omitted] there exists a non-linear infinitely differentiable function f so that [image omitted] for all [image omitted].