Acta mathematica scientia,Series A ›› 2009, Vol. 29 ›› Issue (2): 423-435.

• Articles • Previous Articles     Next Articles

Generalized Bernstein Operators |Defined on a Simplex

 DING Chun-Mei, YANG Ru-Ru   

  1. (Department of Information and Mathematics, |China Jiliang University,  Hangzhou |310018)
  • Received:2007-12-27 Revised:2008-12-07 Online:2009-04-25 Published:2009-04-25
  • Supported by:

    国家自然科学基金(90818020, 60873206)、浙江省自然科学基金(Y7080235)和浙江省教育厅科研重点项目基金(20060543)资助

Abstract:

In this paper, the authors introduce and study  generalized Bernstein operators  defined on a simplex in R.
They first prove that these operators possess the symmetric property for a transformation from the simplex to itself and the Lipshcitz property.  Then,  by means of the multivariate Ditzian-Totik's modulus of continuity, they obtain a strong type direct estimate  and an inverse inequality of weak type for the operators approximating multivariate continuous functions. Finally, they  give some Voronovskaja type expansions when the parameters sn satisfy some conditions. The obtained results include the corresponding ones of the classical Bernstein operators as a corollary.

Key words: Generalized Bernstein operators, Modulus of continuity, Direct theorem, Inverse theorem, Voronovskaja type expansion

CLC Number: 

  • 41A25
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