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

Ostrovska, Sofiya

J. Appl. Math. 2013, Article ID 159720, 7 p. (2013).

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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