Szász-Mirakjan-Durrmeyer and Baskakov-Durrmeyer operators with respect to arbitrary measure

• Elena E. Berdysheva ^[1] ; Eman Al-Aidarous ^[1]
  1. [1] King Abdulaziz University

     King Abdulaziz University

     Arabia Saudí

     
• Localización: Jaen journal on approximation, ISSN 1889-3066, ISSN-e 1989-7251, Vol. 8, Nº. 1, 2016, págs. 1-26
• Idioma: inglés
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• Resumen

  • In this paper, we introduce Sz´asz-Mirakjan-Durrmeyer operators and BaskakovDurrmeyer operators with respect to an arbitrary measure, thus extending this natural generalization from the already studied case of the Bernstein-Durrmeyer operators to these two closely related cases. We establish convergence of the new operators, namely, pointwise convergence at each point of continuity of a function in the support of the measure, and uniform convergence in every compact set in the interior of the support of the measure. In case when the measure is finite, we also show convergence in the corresponding weighted Lp-spaces, 1 ≤ p < ∞. In the latter case, we also give estimates for the rate of convergence in terms of a K-functional.