INVESTIGATION OF THE ASYMPTOTIC BEHAVIOR OF GENERALIZED BASKAKOV-DURRMEYER-STANCU TYPE OPERATORS


Dinlemez Kantar Ü. , Mercan Boyraz M.

EURO ASIA 7th. INTERNATIONAL CONGRESS ON APPLIED SCIENCES, Trabzon, Türkiye, 21 - 22 Ağustos 2020, ss.354-355

  • Basıldığı Şehir: Trabzon
  • Basıldığı Ülke: Türkiye
  • Sayfa Sayıları: ss.354-355

Özet

In[1], generalized Baskakov operators with a constant a≥0 are defined by 𝐵𝑛 𝛼(𝑓, 𝑥) ≔ ∑ 𝑊𝑛,𝑘 𝛼 (𝑥) ∞ 𝑘=0 𝑓 (𝑘 𝑛 ), (1) where 𝑊𝑛,𝑘 𝛼 (𝑥) ∶= 𝑒 −𝑎𝑥 1+𝑥 𝑃𝑘(𝑛,𝑎) 𝑘! 𝑥 𝑘 (1+𝑥)𝑛+𝑘 ,such that ∑ 𝑊𝑛,𝑘 𝛼 (𝑥) ∞ 𝑘=0 = 1 and i ≥ 1 for (n)0 = 1, (n)i : = n. (n + 1) … (n + i − 1) with 𝑃𝑘 (𝑛, 𝑎) ≔ ∑ ( 𝑘 𝑖 ) 𝑘 𝑖=0 (𝑛)𝑖𝑎 𝑘−𝑖 . In[2], for 𝑓 ∈ 𝐶𝐵 [0, ∞), the space of all bounded and continuous functions on [0, ∞), Durrmeyer type of operators in (1) are defined as follows 𝐿𝑛 𝛼(𝑓, 𝑥) ≔ ∑ 𝑊𝑛,𝑘 𝛼 (𝑥) ∞ 𝑘=0 1 𝐵(𝑘 + 1, 𝑛) ∫ 𝑡 𝑘 (1 + 𝑡) 𝑛+𝑘+1 ∞ 0 𝑓(𝑡) 𝑑𝑡 (2) where 𝐵(𝑥, 𝑦) is the Beta function. Let ℒ be all Lebesgue measurable functions 𝑓 on [0, ∞). For 𝑚 positive integer, 𝑓 satisfies ∫ |f(t)| (1+t)m ∞ 0 < ∞. Therefore, In [3], for 𝑓 ∈ ℒ and 𝑛 ∈ 𝑁, the Stancu type generalization of operators (2) is given as follows; 𝐿𝑛,𝑎 𝛼,𝛽(𝑓, 𝑥): = ∑ 𝑊𝑛,𝑘 𝛼 (𝑥) ∞ 𝑘=0 1 𝐵(𝑘 + 1, 𝑛) ∫ 𝑡 𝑘 (1 + 𝑡) 𝑛+𝑘+1 ∞ 0 𝑓 ( 𝑛𝑡 + 𝛼 𝑛 + 𝛽 ) 𝑑𝑡. (3) where 0 ≤ α ≤ β, and 𝑚 < 𝑛. In this presentation, asymptotic approach of operators defined in (3) has been studied with the help of Voronovskaja-type theorem.