The new forms of Voronovskaya's theorem in weighted spaces

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• 作者：Tuncer Acar ; Ali Aral ; Ioan Rasa
• 关键词：Voronovskaya theorem ; Grüss ; type ; Voronovskaya theorem ; Weighted modulus of continuity ; Difference of operators
• 刊名：Positivity
• 出版年：2016
• 出版时间：March 2016
• 年：2016
• 卷：20
• 期：1
• 页码：25-40
• 全文大小：468 KB
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  3.Aral, A., Acar, T.: Voronovskaya type result for \(q\) -derivative of \(q\) -Baskakov operators. J. Appl. Funct. Anal. 7(4), 321–331 (2012)MathSciNet MATH
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• 作者单位：Tuncer Acar (1)
  Ali Aral (1)
  Ioan Rasa (2)
  
  1. Department of Mathematics, Faculty of Science and Arts, Kirikkale University, 71450, Yahsihan, Kirikkale, Turkey
  2. Technical University of Cluj-Napoca, Cluj-Napoca, Romania
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Fourier Analysis
  Operator Theory
  Potential Theory
  Calculus of Variations and Optimal Control
  Econometrics
  
• 出版者：Birkh盲user Basel
• ISSN：1572-9281

文摘

The Voronovskaya theorem which is one of the most important pointwise convergence results in the theory of approximation by linear positive operators (l.p.o) is considered in quantitative form. Most of the results presented in this paper mainly depend on the Taylor’s formula for the functions belonging to weighted spaces. We first obtain an estimate for the remainder of Taylor’s formula and by this estimate we give the Voronovskaya theorem in quantitative form for a class of sequences of l.p.o. The Grüss type approximation theorem and the Grüss-Voronovskaya-type theorem in quantitative form are obtained as well. We also give the Voronovskaya type results for the difference of l.p.o acting on weighted spaces. All results are also given for well-known operators, Szasz-Mirakyan and Baskakov operators as illustrative examples. Our results being Voronovskaya-type either describe the rate of pointwise convergence or present the error of approximation simultaneously.