Yeni bir Schurer tipi Stancu Operatörleri ve ilgili GBS Operatörleri ile Yaklaşım

Öz

Bu çalışma, Schurer tipi Stancu operatörlerinin yeni bir genelleştirmesini sunmakta ve bu operatörlerin yaklaşım özelliklerini incelemektedir. Bu operatörlerin düzgün yakınsaklığı Korovkin Teoremi yardımıyla verilmiş ve yakınsama hızları süreklilik modülü cinsinden ifade edilmiştir. Daha sonra Grüss-Voronovskaja olarak bilinen teorem ispatlanmıştır. Ayrıca, ilgili genelleştirilmiş Boolean toplamı (GBS) operatörleri tanımlanmış ve bu operatörlerin yaklaşım hızları karma düzgünlük modülü ile Lipshitz sınıfından fonksiyonlar kullanılarak elde edilmiştir. Sonrasında, her iki operatör için sayısal örnekler ve grafiksel sonuçlar sunulmuştur.

Anahtar Kelimeler

• Schurer-Stancu operatörleri
• Süreklilik modülü
• Grüss-Voronovskaja tipi teorem
• GBS operatörleri

Approximation by a new Schurer type Stancu Operators and Associated GBS

Öz

This study presents a novel extension of the Schurer type Stancu operators and investigates their properties in terms of approximation. The uniform convergence of these operators is provided using the Korovkin Theorem, and the rates of convergence are expressed in terms of the modulus of continuity. Subsequently, the theorem known as Grüss-Voronovskaja is proven. In addition, the related generalized Boolean sum (GBS) operators are defined, and the rates of approximation for these operators are obtained using the mixed modulus of smoothness and functions from the Lipshitz class. Then, numerical examples and graphical results for both operators are presented.

Anahtar Kelimeler

• Schurer-Stancu operators;
• modulus of continuity;
• Grüss-Voronovskaja type theorem
• GBS operators.

Kaynakça

1. [1] Bernstein, S. N., Demonstration du th'eoreme de Weierstrass fondee sur le calcul des probabilities, Communications of the Kharkov Mathematical Society, 13, 2, 1912.
2. [2] Schurer, F., Linear positive operators in approximation theory, Applied Mathematics Institute Technische Universiteit Delft: report, 1962.
3. [3] Bărbosu, D., Schurer-Stancu type operators, Studia Universitatis Babeş-Bolyai Mathematica, XLVIII, 3(8), 31-35, 2003.
4. [4] Bodur, M., Manav, N., Tasdelen, F., Approximation Properties of λ-Bernstein -Kantorovich-Stancu Operators, Mathematica Slovaca, 72, 1, 2022.
5. [5] Çetin, N., Acu, A. M., Approximation by α–Bernstein–Schurer–Stancu Operators, Journal of Mathematical Inequalities, 15:2, 2021.
6. [6] Vedi, T., Özarslan, M. A., Chlodowsky-type q-Bernstein-Stancu-Kantorovich operators, Journal of Inequalities and Applications, 1, 2015.
7. [7] Stancu, D. D., Quadrature formulas constructed by using certain linear positive operators, Numerical Integration (Proceedings of the Conference, Oberwolfach, 1981), ISNM 241-251, Birkhauser Verlag, Basel, 57, 1982.
8. [8] Stancu, D. D., Approximation of functions by means of a new generalized Bernstein operator, Calcolo, 20: 211–229, 1983.

9. [9] Bostancı, T., Başcanbaz-Tunca, G., On Stancu operators depending on a non -negative integer, Filomat, 36, 18, 2022.
10. [10] Çetin, N., Bascanbaz-Tunca, G., Approximation by a new complex generalized Bernstein operators, Universitatii din Oradea Fascicola Matematica, 26, 2, 2019.
11. [11] Yun, L., Xiang, X., On shape-preserving properties and simultaneous approximation of Stancu operator, Analysis in Theory and Applications, 24, 2008.
12. [12] Devore, R. A., Lorentz, G. G., Constructive Approximation, Grunddlehren der Mathematischen Wissenschaften, Band 303 (Springer-Verlag, Berlin, Heidelberg, New York, and London), 1993.
13. [13] Çetin, N., & Mutlu, N. M., Complex Generalized Stancu-Schurer Operators, Mathematica Slovaca, 74(5), 1215-1232, 2024.
14. [14] Gal, S. G., Gonska, H., Grüss and Grüss-Voronovskaya-type estimates for some Bernstein -type polynomials of real and complex variables, Jaen Journal on Approximation, 7, 1, 2015.
15. [15] Yang, R., Xiong, J., Cao, F., Multivariate Stancu operators defined on a simplex, Applied Mathematics and Computation, 138, 189-198, 2003.
16. [16] Başcanbaz‐Tunca, G., Erençin A., Olgun, A., Quantitative estimates for bivariate Stancu operators, Mathematical Methods in the Applied Sciences, 42 (16), 5241-5250, 2019.
17. [17] Bögel, K., Mehrdimensionale Differentiation von Funktionen mehrerer reeller Veränderlichen, Journal fur die Reine und Angewandte Mathematik, 170, 197-217, 1934.
18. [18] Bögel, K., Über mehrdimensionale Differentiation, Integration und beschränkte Variation, Journal fur die Reine und Angewandte Mathematik, 173, 5-30, 1935.
19. [19] Bögel, K., Über die mehrdimensionale differentiation, Jahresber Deutsch Math-Verein, 65, 45-71, 1962.
20. [20] Badea, C., Cottin, C., Korovkin-type theorems for Generalised Boolean Sum operators, Colloquia Mathematica Societatis Janos Bolyai, Approximation Theory, Kecskemét (Hungary), 58, 51-67, 1990.
21. [21] Badea, C., Badea, I., Cottin, C., Gonska, H., Notes on the degree of approximation of B-continuous and B-differentiable functions, Approximation Theory and Its Applications, 4, 95-108, 1988.
22. [22] Barbosu, D., Some generalized bivariate Berstein operators, Mathematical Notes (Miskolc), 1, 1, 3-10, 2000.
23. [23] Barbosu, D., Bivariate Operators of Schurer-Stancu Type, Analele stiintifice ale Universitatii Ovidius Constanta, 11, 1, 1-8, 2003.
24. [24] Miclauş, D., On the GBS Bernstein–Stancu’s type operators, Creative Mathematics and Informatics, 22, 73-80, 2013.
25. [25] Pop, O. T., Farcas, M. D., Approximation of B-continuous and B-differentiable functions by GBS operators of Bernstein bivariate polynomials, Journal of Inequalities in Pure and Applied Mathematics, 7(3), 92, 2006.
26. [26] Farcaş, M. D., About approximation of B-continuous functions of three variables by GBS operators of Bernstein type on a tetrahedron, Acta Universitatis Apulensis, Mathematics-Informatics, 16, 93-102, 2008.
27. [27] Farcaş, M. D., About approximation of B-continuous and B-differential functions of three variables by GBS operators of Bernstein–Schurer type, Buletinul ştiinţific al Universităţii "Politehnica" din Timişoara. Seria matematică-fizică, 52, 66, 13-22, 2007.
28. [28] Sidharth, M., Ispir, N., Agrawal, P. N., GBS operators of Bernstein–Schurer–Kantorovich type based on q-integers, Applied Mathematics and Computation, 269, 558-568, 2015.
29. [29] Bărbosu, D., Bărbosu, M., On the sequence of GBS operators of Stancu-type, Buletinul științific al Universitatii Baia Mare, Seria B, Fascicola matematică-informatică, 1-6, 2002.
30. [30] Badea, I., Modulus of continuity in Bögel sense and some applications for approximation by a Bernstein-type operator, Studia Universitatis Babes-Bolyai, Series Mathematica-Mechanica (Romanian), 18, 2, 69-78, 1973.