Parçalı Bir Fonksiyonla Tanımlı Bernstein Stancu Operatörleri

Öz

Bu çalışmada Stancu tipi operatörlerin parametreye bağlı iki farklı genelleştirmesinin yaklaşım özellikleri incelenmiştir. İlk aşamada, [-1,1] aralığında tanımlı bu operatörün teoremi sağlayan Korovkin tipi bir operatör olduğu belirlenmiş ve önemli özellikleri incelenmiştir. Daha sonra, bu operatör kullanılarak Kantorovich tipi yeni bir operatör sınıfı tanımlanmış ve bu operatörlerin yaklaşım özellikleri üzerine çalışılmıştır. Çalışmanın bir diğer önemli kısmı ise her iki sınıf operatörün Lp uzaylarında yakınsaklık özelliklerinin incelenmesidir. Bu bağlamda, operatörlerin fonksiyonlar üzerindeki etkisi ve yakınsaklık özellikleri değerlendirilmiş ve yeni tanımlanan operatörlerin klasik yaklaşımlara göre avantajları gösterilmiştir. Ayrıca, bu operatörlerin yaklaşım grafikleri sunulmuş ve operatörlerin fonksiyonlar üzerindeki etkileri görsel olarak analiz edilmiştir. Çalışma, her iki operatörün teorik analizini ve görsel sonuçlarını sunarak, yakınsamaları hakkında önemli bilgiler sağlamaktadır.

Anahtar Kelimeler

• Bernstein Stancu Operatörleri
• Korovkin Teoremi
• Kantorovich Operatörleri

Stancu and Kantorovich-Type Generalizations of a Bernstein Operator: Approximating Locally Integrable Functions

Öz

This study investigates the approximation properties of two different types of parameter-dependent generalizations of Stancu type operators. In the first step, it is determined that this operator defined on the interval [-1,1] is an operator of Korovkin type satisfying the theorem and its important properties are analyzed. Then, a new class of operators of Kantorovich type is defined using this operator and the study on the approximation properties of these operators is elaborated. Another important part of the study is to investigate the convergence properties of both classes of operators in Lp spaces. In this context, the effect of the operators on functions and their convergence properties are evaluated and the advantages of the newly defined operators over the classical approximations are demonstrated. In addition, graphs of the approximation of these operators are presented and the effects of the operators on the functions are visually analyzed. By presenting the theoretical analysis and visual results of both operators, the study provides important information about their convergence.

Anahtar Kelimeler

• Bernstein Stancu Operators
• Kantorovich Operators
• Korovkin's Theorem

Kaynakça

1. [1] Weierstrass, K., Über die analytische Darstellbarkeit sogenannter willkürlicher Functionen einer reellen Veränderlichen. Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin, 2, 633-639, 1885.
2. [2] Bernstein, S., Démoistration du théorème de Weierstrass fondée sur le calcul des probabilités. Сообщенiя Харьковскаго математическаго общества, 13(1), 1-2, 1912.
3. [3] Stancu, D.D., On Some Polynomials of Bernstein Type, Studii și Cercetări Științifice, Seria Matematică (Iaşi), 11, 221-233, 1960.
4. [4] Stancu, D.D., Approximation of functions by a new class of polynomial operators, Revue Roumaine de Mathématiques Pures et Appliquées, 13(8), 1173-1194, 1968.
5. [5] Stancu, D.D., Quadrature formulas constructed by using certain linear positive operators, In Numerical Integration: Proceedings of the Conference Held at the Mathematisches Forschungsinstitut Oberwolfach, October 4–10, 1981, 241-25, Birkhäuser, Basel, 1982.
6. [6] Stancu, D.D., Occorsio, M.R., On approximation by binomial operators of Tiberiu Popoviciu type, Revue d'analyse numérique et de théorie de l'approximation, 27(1), 167-181, 1998.
7. [7] Stancu, D.D., A note on a multiparameter Bernstein-type approximating operator, Mathematica (Cluj), 26(49), 153-157, 1984.
8. [8] Stancu, D.D., Approximation of functions by means of a new generalized Bernstein operator, Calcolo, 20, 211-229, 1983.

9. [9] Stancu, D.D., Asupra unei generalizări a polinoamelor lui Bernstein, Studia Universitatis Babeş-Bolyai, Mathematica, 14(2), 31-45, 1969.
10. [10] Stancu, D.D., The remainder in the approximation by a generalized Bernstein Operator: a representation by a convex combination of second-order divided differences, Calcolo, 35(1), 53-62, 1998.
11. [11] Kantorovich, L.V., Sur certains développements suivant les polynômes de la forme S. Bernstein, Comptes Rendus de l’Académie des Sciences de l’URSS, 20, 563–568, 1930.
12. [12] Korovkin, P.P., On convergence of linear positive operators in the space of continuous functions, Doklady Akademii Nauk SSSR, 90, 961–964, 1953.
13. [13] Korovkin, P.P., Linear operators and approximation theory, Hindustan Publishing Corporation, 1960.
14. [14] Campiti, M., Metafune, G., L^p-convergence of Bernstein-Kantorovich-type operators, Annales Polonici Mathematici, 63(3), 273-280, 1996.
15. [15] Altomare, F., Campiti, M., Korovkin-type approximaton theory and its applications, Walter de Gruyter, 1994.
16. [16] Altomare, F., Korovkin-type theorems and approximation by positive linear operators, arXiv:1009.2601, 2010.
17. [17] Altomare, F., Cappelletti Montano, M., Leonessa, V., Raşa, I., A generalization of Kantorovich operators for convex compact subsets, Banach Journal of Mathematical Analysis, 11(3), 591-614, 2017.
18. [18] Kajla, A., Mursaleen, M., Acar, T, Durrmeyer-type generalization of parametric Bernstein operators, Symmetry, 12(7), 1141, 2020.
19. [19] Kahvecibaşı, İ., Approximation properties of Bernstein–Kantorovich operators on the interval [-1,1], Harran University, Şanlıurfa, Türkiye, 2014.
20. [20] Kajla, A., The Kantorovich variant of an operator defined by D.D. Stancu, Applied Mathematics and Computation, 316, 400-408, 2018.
21. [21] Stancu, D.D., Approximation of functions by a new class of linear polynomial operators, Revue Roumaine de Mathématiques Pures et Appliquées, 13(8), 1173–1194, 1968.
22. [22] Kantorovich, L.V., Akilov, G.P., Functional analysis, Pergamon Press, 1st ed., 1982.
23. [23] Sasi, S., Jyothi, L.S., A heuristic approach for secured transmission of image based on Bernstein polynomial. In International Conference on Circuits, Communication, Control and Computing, 312-315, 2014.
24. [24] Bilgin, N.G., Kaya, Y., and Eren, M., Security of image transfer and innovative results for (p, q)-Bernstein-Schurer operators, AIMS Mathematics, 9(9), 23812-23836, 2024.
25. [25] Çilo, A. Ural, A., İzgi, A., Comparison of Bernstein polynomials and some modifications, In XXV. National Mathematics Symposium - Niğde University, 5-8 September, 2012.
26. [26] Çilo, A., Approximation properties and rates of convergence of Bernstein polynomials on the interval [-1,1], Harran University, Şanlıurfa, Türkiye, 2012.
27. [27] İzgi, A. ve Büyükyazıcı, İ., Approximation in boundedness interval and order of approximation, Kastamonu Education Journal, 11(2), 451-460, 2003.
28. [28] İzgi, A., Approximation by a class of new type Bernstein polynomials of one and two variables, Global Journal of Pure and Applied Mathematics, 8(5), 55-71, 2012.
29. [29] Büyükyazici, İ., Bernstein polynomials of functions of two variables, Ankara University, Ankara, Türkiye, 1999.
30. [30] Bilgin, N.G., Eren, M., A generalization of two dimensional Bernstein-Stancu operators, Sinop University Journal of Natural Sciences, 6(2), 130-142, 2021.
31. [31] Lorentz, G. G., Bernstein Polynomials. University of Toronto Press, 1953.
32. [32] Tunç, T. and Alhazzori, G., On conic equations under Bernstein operators, Bitlis Eren University Journal of Science, 13(1), 161-169, 2024.
33. [33] Aral, A., Ökten, O., Acar, T., A note on Bernsteın-Stancu-Chlodowsky operators, Kırıkkale Üniversitesi Bilimde Gelismeler Dergisi, 11, 2012.
34. [34] Aral, A., Cardenas-Morales, D., Garrancho, P., Bernstein-type operators that reproduce exponential functions, Journal of Mathematical Inequalities, 12(3), 861-872, 2018.
35. [35] Güney, B.T., Generalized Stancu Operators. Ankara University, Ankara, Türkiye, 2022.
36. [36] Abramowitz, M., Stegun I.A., Handbook of mathematical functions with formulas, graphs, and mathematical tables, US Government Printing Office, 1965.
37. [37] Hacısalihoğlu, H., Hacıyev, A., Convergence of sequences of linear positive operators, Ankara University, Ankara, Türkiye, 1-30, 1995.
38. [38] Bostancı, T., Başcanbaz-Tunca, G., On Stancu operators depending on a non-negative integer, Filomat, 36(18), 6129-6138, 2022.
39. [39] Kaytmaz, Ö., The problem of determining source term in a kinetic equation in an unbounded domain, AIMS Mathematics, 9(4), 9184-9194, 2024.
40. [40] Bilgin, N.G., Coşkun, N., Comparison result of some Gadjiev-Ibragimov type operators, Karaelmas Fen ve Mühendislik Dergisi, 8(1), 188-196, 2018.
41. [41] Aslan, R., Some approximation results on λ Szasz-Mirakjan-Kantorovich operators, Fundamental Journal of Mathematics and Applications, 4(3), 150-158, 2021.
42. [42] Çiçek, H., İzgi, A., Approximation by modified bivariate Bernstein-Durrmeyer and GBS bivariate Bernstein-Durrmeyer operators on a triangular region, Fundamental Journal of Mathematics and Applications, 5(2), 135-144, 2022.
43. [43] Baytunç, E., Aktuğlu, H., Mahmudov, N., A new generalization of Szász-Mirakjan Kantorovich operators for better error estimation, Fundamental Journal of Mathematics and Applications, 6(4), 194-210, 2023.
44. [44] Arı, D.A., Yılmaz, G.U., A note on Kantorovich type operators which preserve affine functions, Fundamental Journal of Mathematics and Applications, 7(1), 53-58, 2024.
45. [45] Yılmaz, Ö.G., On the eigenstructure of the q-Stancu operator, Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 73(3), 820-832, 2024.
46. [46] Bodur, M., Bostancı, T., Başcanbaz-Tunca, G., On Kantorovich variant of Brass-Stancu operators, Demonstratio Mathematica, 57(1), 20240007, 2024.
47. [47] Kumar, A., A new kind of variant of the Kantorovich type modification operators introduced by D.D. Stancu, Results in Applied Mathematics, 11, 100158, 2021.
48. [48] Yeter, S., Çetin, N., Approximation by generalized Stancu-Kantorovich operators, Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 74(3), 503-512, 2025.
49. [49] Başcanbaz-Tunca, G., A note on Stancu operators with three parameters, Bulletin of the Transilvania University of Brasov. Series III: Mathematics and Computer Science, 5(67), 55-70, 2025.
50. [50] Çetin, N., Manav Mutlu, N., Approximation by Generalization of Bernstein–Schurer Operators. In Advanced mathematics for the modeling and solution of challenging problems in engineering, 582, 375-391, 2025.