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Journals / Erciyes Üniversitesi Fen Bilimleri Enstitüsü Dergisi / 2020 / Cilt: 36 - Sayı: 1

Approximation By 𝒒 −Szász-Kantorovich-Chlodowsky Operators On Weighted Spaces

Ağırlıklı Uzaylarda 𝒒 −Szász-Kantorovich-Chlodowsky Operatörlerinin Yaklaşımlar

Aydın İZGİ
Reşat ASLAN
Pages
138–150
Publication Date
2022-10-18
DOI
—
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AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

Finally, the rate of convergence in weighted spaces is calculated with the help of modulus of continuity.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:The main purpose of this research is to apply 𝑞-calculus to some linear positive operators and to estimate the approximation properties of these operators, which is a popular area of study in recent years of the approximation theory which has a significant contribution to mathematics in the last two centuries and has many applications.
📊
Key Findings & Data:Firstly some definitions and notations related to 𝑞-calculus are given, then the images of the test functions 1,𝑡,𝑡 2 ,𝑡 3 ,𝑡 4 under 𝑞-Szász-Kantorovich-Chlodowsky operators are calculated and the central moments of the 𝑞 -Szász-Kantorovich-Chlodowsky operators up to 4-th order are given.
💡
Conclusion & Impact:Finally, the rate of convergence in weighted spaces is calculated with the help of modulus of continuity.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Domain Parameters

Abstract

The main purpose of this research is to apply 𝑞-calculus to some linear positive operators and to estimate the approximation properties of these operators, which is a popular area of study in recent years of the approximation theory which has a significant contribution to mathematics in the last two centuries and has many applications. Firstly some definitions and notations related to 𝑞-calculus are given, then the images of the test functions 1,𝑡,𝑡 2 ,𝑡 3 ,𝑡 4 under 𝑞-Szász-Kantorovich-Chlodowsky operators are calculated and the central moments of the 𝑞 -Szász-Kantorovich-Chlodowsky operators up to 4-th order are given. In addition, theorems related to the uniform convergence of the above operator in the weighted spaces and to obtain the degree of approximation of this operator the proof of the Voronovskaja type asymptotic theorem are given. Finally, the rate of convergence in weighted spaces is calculated with the help of modulus of continuity.

Özet

Bu araştırmanın asıl amacı, son iki yüzyıla girildiğinde matematiğeönemli katkısı olan ve birçok uygulama alanına sahip olan yaklaşımlarteorisinde son yılların popüler bir çalışma alanlarından 𝑞-calculus’ün bazılineer pozitif operatörlere uygulanması ve bu operatörlerin yaklaşımözelliklerinin tahmin edilmesi üzerine olacaktır. Öncelikle 𝑞-calculus ile ilgilibazı tanım ve notasyonlar verilmiş ardından 1,𝑡,𝑡2,𝑡3,𝑡4testfonksiyonlarının 𝑞-Szász-Kantorovich-Chlodowsky operatörleri altındakigörüntüleri hesaplanmış ve 𝑞-Szász-Kantorovich-Chlodowsky operatörünün4. mertebeye kadar olan merkezi momentleri verilmiştir. Ayrıca yukarıdakioperatörün ağırlıklı uzaylardaki düzgün yakınsaklığı ile ilgili teoremler ve buoperatörün yaklaşım derecesini elde etmek için Voronovskaja tip asimtotikteoremin ispatı verilmiştir. Son olarak süreklilik modülü yardımıyla ağırlıklıuzaylardaki yakınsaklık oranı hesaplanmıştır.

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Cite This Article

İZGİ, A., ASLAN, R. (2022). Approximation By 𝒒 −Szász-Kantorovich-Chlodowsky Operators On Weighted Spaces. Erciyes Üniversitesi Fen Bilimleri Enstitüsü Dergisi, Cilt: 36 - Sayı: 1, 138–150. https://ajindex.com/articles/503236

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References (19)

  1. Pinkus, A. 2000. Weierstrass and Approximation Theory. J. Approx Theory, 107:1-66
  2. Bernstein, S. 1912. Démonstration du Théorémé de Weierstrass, Fondee Surla Calcul Des Probabilities. Commun. Soc. Math. Kharkow,13(2): 1-2
  3. Kantorovich, L. V. 1930. Sur Certain Developpements Suivant les Polynomes de La Forme de S. Bernstein, I, II, C.R. Acad. URSS, 563-568, 595-600
  4. Chlodovsky, I. 1937. Sur le Developpement Des Fonctions Definies Dans Un Intervalle Infini en Series De Polynomes de M. S. Bernstein. Compos Math,4:380-393
  5. Szász, O. 1950. Generalization of S.Bernstein’s Polynomials to Infinite Interval, J. Research Nat. Bur. Standarts, 45, 239-245
  6. Bohman, H. 1952. On Approximation of Continuous and of Analytic Functions. Arkiv för Matematik, 2(1), 43-56
  7. Korovkin, P. P. 1953. On Convergence of Linear Positive Operators In The Space of Continuous Functions. Dokl Akad Nauk SSSR, 90:961-964
  8. Lupaş, L., Lupaş, A. 1987. Polynomials of Binomial Type and Approximation Operators. Studia Univ. Babes-Bolyai, Mathematica, 32(4), 61-69
  9. Phillips, G. M. 1997. Bernstein Polynomials based on the q−integers, Ann. Numer. Math. 4, no. 1- 4,511-518
  10. Oruç, H., Tuncer, N. 2002. On the Convergence and Iterates of q-Bernstein Polynomials. Journal of Approximation Theory, 117(2), 301-313
  11. Videnskii, V. S. 2005. On Some Classes of q-Parametric Positive Linear Operators. In Selected Topics in Complex Analysis (pp. 213-222). Birkhäuser Basel
  12. Aral, A., Gupta, V. 2006. The q -derivative and Applications to q -Szász Mirakyan Operators. Calcolo 43, no. 3, 151–170
  13. Karsli, H., Gupta, V. 2008. Some Approximation Properties of q-Chlodowsky Operators. Applied Mathematics and Computation, 195(1), 220-229
  14. Büyükyazıcı, İ. 2010. Approximation By Stancu–Chlodowsky Polynomials. Computers & mathematics With Applications, 59(1), 274-282
  15. Dalmanoglu, Ö., Dogru, O. 2010. On Statistical Approximation Properties of Kantorovich Type qBernstein operators. Mathematical and Computer Modelling, 52(5-6), 760-771
  16. Aral, A. 2008. A Generalization of Szász–Mirakyan Operators Based on q-integers. Mathematical and Computer Modelling, 47(9-10), 1052-1062
  17. Karaisa, A., Aral, A. 2016. Some Approximation Properties of Kantorovich Variant of Chlodowsky Operators Based on 𝑞-integer, 65,97-119 Ankara
  18. İspir, N. 2001. On Modified Baskakov Operators on Weighted Spaces. Turk. J. Math. 25:355- 365
  19. Gadjiev, A. D. 1976. Theorems of the Typ