Bu çalışmada üstel fonksiyonları yeniden üreten Bernstein-Stancu-Kantorovich operatörlerinin bir genellemesi sunulmuştur. Uygun fonksiyon uzayları için hem düzgün hem de L^p yakınsaması kurulmuştur. Yeni operatörlerin üstel fonksiyonu sağladığını kanıtladık ve iyi bilinen Korovkin Teoremini kullanarak düzgün yakınsaklık sonucunu elde etmek için gerekli olan e ^μxin çeşitli kuvvetlerine göre değerlendirilen operatörlerin analitik ifadelerini hesapladık. Sonuç olarak L_μ^p ([0,1]) ağırlıklı uzayını kendisine aktaran yeni operatörler için yakınsama teoremi kurulmuştur. Ek olarak, sürekli durumda tahmin edilen fonksiyonun olağan süreklilik modülünü kullanarak, yaklaşık hatası için niceliksel tahminler verilmiştir.
Bernstein, S. N., Demonstration du theoreme de weierstrass fondee sur le calcul de probabilities, Commun. Soc. Math. Kharkow, 2, 1–2, (1912– 1913).
Stancu, D. D., Approximation of function by a new class of polynomial operators, Rev. Roum. Math. Pures et Appl., 13, 8, 1173–1194, (1968).
Morigi, S., Neamtu, M., Some results for a class of generalized polynomials, Adv. Comput. Math., 12, 133–149, (2000).
Aral, A., Cardenas-Morales, D., Garrancho, P., Bernstein-type operators that reproduce exponential functions, J. Math. Inequal., 3, 861–872, (2018).
Angeloni, L., Costarelli, D., Approximation by exponential-type polynomials, Journal of Mathematical Analysis and Applications, 532, 1, 127927 (2024).
Barbosu, D., Kantorovich-Stancu type operators, Journal of Inequalities in Pure and Applied Mathematics, 5, 3, (2004).
Altomare, F., Campiti, M., Korovkin-Type Approximation Theory and Its Applications, Walter de Gruyter, Berlin, (1994).
Altomare, F., Korovkin-type theorems and approximation by positive linear operators, arXiv, https://doi.org/10.48550/arXiv.1009.2601, (2010).
Paşca, S. V., The modified Bernstein-Stancu operators, General Mathematics, 291, 121-128, (2021).
Acar, E., Izgi, A., Kırcı Serenbay, S., Note On Jakimovski-Leviatan Operators Preserving ex, Applied Mathematics and Nonlinear Sciences, 4 2, 543–550, (2019).
Acar, E., Özalp Güller, Ö., Kırcı Serenbay, S., Approximation by non-linear Meyer-König and Zeller operators based on q-integers, International Journal of Mathematics and Computer in Engineering, 2, 2, 71–82, (2024).
Acar, E., Kırcı Serenbay, S., Approximation by Nonlinear q-Bernstein- Chlodowsky Operators, TWMS J. App. and Eng. Math., 14, 1, 42–51, (2024).
Acar, E., Holhoş, A., Kırcı Serenbay, S., Polynomial Weighted Approximation by Szasz-Mirakyan Operators of Max-product Type, Kragujevac Journal of Mathematics, 49, 3, 365–373 (2025).
Acar, E., Izgi, A., On Approximation by Generalized Bernstein-Durrmeyer Operators, J. Adv. Math. Stud., 14, 3, 352–361, (2021).
Aral, A., Aydın Arı, D., Yılmaz, B., A Note on Kantorovich Type Bernstein Chlodowsky Operators Which Preserve Exponential Function, Journal of Mathematical Inequalities, 15, 3, 1173–1183, (2021).
Acu, A. M., Aral, A., Rasa, I., New properties of operators preserving exponentials, Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 117 (2023).
On approximation properties by exponential type of Bernstein-Stancu Operators
In the paper, we introduced a generalization of Bernstein-Stancu-Kantorovich operators that reproduces exponential functions. For appropriate function spaces, both the uniform and L^p convergence have been established. We proved that the new operators satisfy the Korovkin tests with the exponential functions and calculated the operators’ analytical expressions evaluated on various powers of e ^μxwhich is necessary to get the uniform convergence conclusion using the well-known Korovkin Theorem. Consequently, the convergence theorem for the new operators, which transfer the weighted space L_μ^p ([0,1]) to itself, has been established. Additionally, using the usual modulus of continuity of the estimated function in the continuous case, we provide quantitative estimates for the approximation error.
Bernstein, S. N., Demonstration du theoreme de weierstrass fondee sur le calcul de probabilities, Commun. Soc. Math. Kharkow, 2, 1–2, (1912– 1913).
Stancu, D. D., Approximation of function by a new class of polynomial operators, Rev. Roum. Math. Pures et Appl., 13, 8, 1173–1194, (1968).
Morigi, S., Neamtu, M., Some results for a class of generalized polynomials, Adv. Comput. Math., 12, 133–149, (2000).
Aral, A., Cardenas-Morales, D., Garrancho, P., Bernstein-type operators that reproduce exponential functions, J. Math. Inequal., 3, 861–872, (2018).
Angeloni, L., Costarelli, D., Approximation by exponential-type polynomials, Journal of Mathematical Analysis and Applications, 532, 1, 127927 (2024).
Barbosu, D., Kantorovich-Stancu type operators, Journal of Inequalities in Pure and Applied Mathematics, 5, 3, (2004).
Altomare, F., Campiti, M., Korovkin-Type Approximation Theory and Its Applications, Walter de Gruyter, Berlin, (1994).
Altomare, F., Korovkin-type theorems and approximation by positive linear operators, arXiv, https://doi.org/10.48550/arXiv.1009.2601, (2010).
Paşca, S. V., The modified Bernstein-Stancu operators, General Mathematics, 291, 121-128, (2021).
Acar, E., Izgi, A., Kırcı Serenbay, S., Note On Jakimovski-Leviatan Operators Preserving ex, Applied Mathematics and Nonlinear Sciences, 4 2, 543–550, (2019).
Acar, E., Özalp Güller, Ö., Kırcı Serenbay, S., Approximation by non-linear Meyer-König and Zeller operators based on q-integers, International Journal of Mathematics and Computer in Engineering, 2, 2, 71–82, (2024).
Acar, E., Kırcı Serenbay, S., Approximation by Nonlinear q-Bernstein- Chlodowsky Operators, TWMS J. App. and Eng. Math., 14, 1, 42–51, (2024).
Acar, E., Holhoş, A., Kırcı Serenbay, S., Polynomial Weighted Approximation by Szasz-Mirakyan Operators of Max-product Type, Kragujevac Journal of Mathematics, 49, 3, 365–373 (2025).
Acar, E., Izgi, A., On Approximation by Generalized Bernstein-Durrmeyer Operators, J. Adv. Math. Stud., 14, 3, 352–361, (2021).
Aral, A., Aydın Arı, D., Yılmaz, B., A Note on Kantorovich Type Bernstein Chlodowsky Operators Which Preserve Exponential Function, Journal of Mathematical Inequalities, 15, 3, 1173–1183, (2021).
Acu, A. M., Aral, A., Rasa, I., New properties of operators preserving exponentials, Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 117 (2023).
Acar, E. (2025). On approximation properties by exponential type of Bernstein-Stancu Operators. Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 27(1), 315-323. https://doi.org/10.25092/baunfbed.1553994
AMA
Acar E. On approximation properties by exponential type of Bernstein-Stancu Operators. BAUN Fen. Bil. Enst. Dergisi. Ocak 2025;27(1):315-323. doi:10.25092/baunfbed.1553994
Chicago
Acar, Ecem. “On Approximation Properties by Exponential Type of Bernstein-Stancu Operators”. Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi 27, sy. 1 (Ocak 2025): 315-23. https://doi.org/10.25092/baunfbed.1553994.
EndNote
Acar E (01 Ocak 2025) On approximation properties by exponential type of Bernstein-Stancu Operators. Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi 27 1 315–323.
IEEE
E. Acar, “On approximation properties by exponential type of Bernstein-Stancu Operators”, BAUN Fen. Bil. Enst. Dergisi, c. 27, sy. 1, ss. 315–323, 2025, doi: 10.25092/baunfbed.1553994.
ISNAD
Acar, Ecem. “On Approximation Properties by Exponential Type of Bernstein-Stancu Operators”. Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi 27/1 (Ocak 2025), 315-323. https://doi.org/10.25092/baunfbed.1553994.
JAMA
Acar E. On approximation properties by exponential type of Bernstein-Stancu Operators. BAUN Fen. Bil. Enst. Dergisi. 2025;27:315–323.
MLA
Acar, Ecem. “On Approximation Properties by Exponential Type of Bernstein-Stancu Operators”. Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi, c. 27, sy. 1, 2025, ss. 315-23, doi:10.25092/baunfbed.1553994.
Vancouver
Acar E. On approximation properties by exponential type of Bernstein-Stancu Operators. BAUN Fen. Bil. Enst. Dergisi. 2025;27(1):315-23.