A Survey of Results on the Limit q-Bernstein Operator. (28th March 2013)

• Record Type: Journal Article
• Authors: Ostrovska, Sofiya
• Other Names: Gupta Vijay Academic Editor.
• Is Part Of: Journal of applied mathematics. Volume 2013(2013)
• Publication Date: 2013-03-28
• Subjects: Mathematics -- Periodicals; 519.05
• DOI: 10.1155/2013/159720
• Languages: English
• ISSNs: 1110-757X

Abstract

Abstract : 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.