Abstract
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.
References
- 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).
Abstract
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.
References
- 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).
Details
| Primary Language | English |
|---|---|
| Subjects | Operator Algebras and Functional Analysis |
| Journal Section | Research Articles |
| Authors | |
| Early Pub Date | January 16, 2025 |
| Publication Date | January 20, 2025 |
| Submission Date | September 21, 2024 |
| Acceptance Date | December 27, 2024 |
| Published in Issue | Year 2025 Volume: 27 Issue: 1 |