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**A survey of results on the limit \(q\)-Bernstein operator.**
*(English)*
Zbl 1268.81094

Summary: 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.

### MSC:

81R50 | Quantum groups and related algebraic methods applied to problems in quantum theory |

17B37 | Quantum groups (quantized enveloping algebras) and related deformations |

81R30 | Coherent states |

41A17 | Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities) |

14F10 | Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials |

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\textit{S. Ostrovska}, J. Appl. Math. 2013, Article ID 159720, 7 p. (2013; Zbl 1268.81094)

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### References:

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