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Functions whose smoothness is not improved under the limit q-Bernstein operator

Authors
  • Ostrovska, Sofiya1
  • 1 Atilim University, Department of Mathematics, Ankara, Turkey , Ankara (Turkey)
Type
Published Article
Journal
Journal of Inequalities and Applications
Publisher
Springer International Publishing
Publication Date
Dec 14, 2012
Volume
2012
Issue
1
Identifiers
DOI: 10.1186/1029-242X-2012-297
Source
Springer Nature
Keywords
License
Green

Abstract

The limit q-Bernstein operator Bq emerges naturally as a modification of the Szász-Mirakyan operator related to the Euler probability distribution. At the same time, this operator serves as the limit for a sequence of the q-Bernstein polynomials with 0<q<1. Over the past years, the limit q-Bernstein operator has been studied widely from different perspectives. Its approximation, spectral, and functional-analytic properties, probabilistic interpretation, the behavior of iterates, and the impact on the analytic characteristics of functions have been examined. It has been proved that under a certain regularity condition, Bq improves the smoothness of a function which does not satisfy the Hölder condition. The purpose of this paper is to exhibit ‘exceptional’ functions whose smoothness is not improved under the limit q-Bernstein operator. MSC:26A15, 26A16, 41A36.

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