The unicity theorems for the limit q-Bernstein operator

Abstract

The limit q-Bernstein operator emerges naturally as a q-version of the Szász–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 is a positive shape-preserving linear operator on with 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 for then f is a linear function. The result is sharp in the following sense: for any proper closed subset of [0, 1] satisfying there exists a non-linear infinitely differentiable function f so that for all

Keywords:

• limit q-Bernstein operator
• Szasz–Mirakyan operator
• q-deformed Poisson distribution
• Euler distribution
• analytic function

AMS Subject Classifications::

• 30A99
• 41A36
• 60E05

Acknowledgements

I would like to express my sincere gratitude to Dr Yuri Lyubarsky from the Ben-Gurion University of the Negev for his valuable comments and to P. Danesh from Atilim University Departmental English Language Studies Unit for his assistance in the preparation of the manuscript.