A Survey of Results on the Limit <i>q</I>-bernstein Operator

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2013

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Hindawi Ltd

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Mathematics
(2000)
The Atılım University Department of Mathematics was founded in 2000 and it offers education in English. The Department offers students the opportunity to obtain a certificate in Mathematical Finance or Cryptography, aside from their undergraduate diploma. Our students may obtain a diploma secondary to their diploma in Mathematics with the Double-Major Program; as well as a certificate in their minor alongside their diploma in Mathematics through the Minor Program. Our graduates may pursue a career in academics at universities, as well as be hired in sectors such as finance, education, banking, and informatics. Our Department has been accredited by the evaluation and accreditation organization FEDEK for a duration of 5 years (until September 30th, 2025), the maximum FEDEK accreditation period achievable. Our Department is globally and nationally among the leading Mathematics departments with a program that suits international standards and a qualified academic staff; even more so for the last five years with our rankings in the field rankings of URAP, THE, USNEWS and WEBOFMETRIC.

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The limit q-Bernstein operator B-q emerges naturally as a modification of the Szasz-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. is a positive shape-preserving linear operator on C[0, 1] with parallel to B-q parallel to = 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.

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