Research Article
Year 2025,
, 958 - 971, 24.06.2025
Abstract
In this paper, we get new Voronovskaja-type asymptotic formulae for modified Bernstein operators by using regular summability methods. We also display some significant special cases of our results including the methods of Cesàro summability, Riesz summability, Abel summability and Borel summability. At the end, we also discuss the similar results for the Kantorovich version of the operators.
Keywords
Bernstein polynomials Bernstein-Kantorovich polynomials Voronovskaja-type asymptotic formula regular summability methods matrix methods power series methods
References
- [1] M. E. Alemdar and O. Duman, General summability methods in the approximation by Bernstein-Chlodovsky operators, Numer. Funct. Anal. Optim. 42 (5), 497–509, 2021.
- [2] I. Aslan and O. Duman, Approximation by nonlinear integral operators via summability process, Math. Nachr. 293 (3), 430–448, 2020.
- [3] O. G. Atlihan and C. Orhan, Summation process of positive linear operators, Comput. Math. Appl. 56 (5), 1188–1195, 2008.
- [4] S. Bernstein, Démonstration du théorème de Weierstrass fondée sur le calcul des probabilités (Proof of the theorem of Weierstrass based on the calculus of probabilities), Comm. Kharkov Math. Soc. 13, 1–2, 1912.
- [5] R. Bojanić and F. H. Chêng, Estimates for the rate of approximation of functions of bounded variation by Hermite-Fejér polynomials, Second Edmonton Conference on Approximation Theory (Edmonton, Alta., 1982), CMS Conf. Proc., 3, 5–17, 1983.
- [6] J. Boos, Classical and Modern Methods in Summability, Oxford University Press, Oxford, 2000.
- [7] D. F. Dawson, Matrix summability over certain classes of sequences ordered with respect to rate of convergence, Pacific J. Math. 24, 51–56, 1968.
- [8] K. Demirci, S. Yildiz and F. Dirik, Approximation via power series method in twodimensional weighted spaces, Bull. Malays. Math. Sci. Soc. 43 (6), 3871–3883, 2020.
- [9] L. Fejér, Untersuchungen über Fouriersche Reihen, Math. Annalen 58, 51–69, 1904.
- [10] T. Y. Gokcer and O. Duman, Regular summability methods in the approximation by max-min operators, Fuzzy Sets and Systems, 426, 106–120, 2022.
- [11] G. H. Hardy, Divergent Series, Oxford, at the Clarendon Press, 1949.
- [12] L. V. Kantorovich, Sur certains développements suivant les polynômes de la forme de S. Bernstein, I. C. R. Acad. Sci. URSS , 563–568, 1930.
- [13] T. A. Keagy and W. F. Ford, Acceleration by subsequence transformations, Pacific J. Math. 132 (2), 357–362, 1988.
- [14] G. G. Lorentz, Bernstein Polynomials, Mathematical Expositions, No. 8. University of Toronto Press, Toronto, 1953.
- [15] R. N. Mohapatra, Quantitative results on almost convergence of a sequence of positive linear operators, J. Approx. Theory 20 (3), 239–250, 1977.
- [16] G. M. Phillips, Bernstein polynomials based on the q-integers, Ann. Numer. Math. 4 (1-4), 511–518, 1997.
- [17] I. Sakaoglu and C. Orhan, Strong summation process in Lp spaces, Nonlinear Anal. 86, 89–94, 2013.
- [18] D. A. Smith and W. F. Ford, Acceleration of linear and logarithmic convergence, SIAM J. Numer. Anal. 16 (2), 223–240, 1979.
- [19] E. Tas and T. Yurdakadim, Approximation by positive linear operators in modular spaces by power series method, Positivity, 21 (4), 1293–1306, 2017.
- [20] M. Unver and C. Orhan, Statistical convergence with respect to power series methods and applications to approximation theory, Numer. Funct. Anal. Optim. 40 (5), 535– 547, 2019.
- [21] E. Voronovskaja, Détermination de la forme asymptotique d’approximation des fonctions par les polynômes de M. Bernstein, Dokl. Akad. Nauk SSSR, 79–85, 1932.
- [22] J. Wimp, Sequence Transformations and Their Applications, Math. Sci. Eng. Vol. 154, Academic Press, New York-London, 1981.
Year 2025,
, 958 - 971, 24.06.2025
Abstract
References
- [1] M. E. Alemdar and O. Duman, General summability methods in the approximation by Bernstein-Chlodovsky operators, Numer. Funct. Anal. Optim. 42 (5), 497–509, 2021.
- [2] I. Aslan and O. Duman, Approximation by nonlinear integral operators via summability process, Math. Nachr. 293 (3), 430–448, 2020.
- [3] O. G. Atlihan and C. Orhan, Summation process of positive linear operators, Comput. Math. Appl. 56 (5), 1188–1195, 2008.
- [4] S. Bernstein, Démonstration du théorème de Weierstrass fondée sur le calcul des probabilités (Proof of the theorem of Weierstrass based on the calculus of probabilities), Comm. Kharkov Math. Soc. 13, 1–2, 1912.
- [5] R. Bojanić and F. H. Chêng, Estimates for the rate of approximation of functions of bounded variation by Hermite-Fejér polynomials, Second Edmonton Conference on Approximation Theory (Edmonton, Alta., 1982), CMS Conf. Proc., 3, 5–17, 1983.
- [6] J. Boos, Classical and Modern Methods in Summability, Oxford University Press, Oxford, 2000.
- [7] D. F. Dawson, Matrix summability over certain classes of sequences ordered with respect to rate of convergence, Pacific J. Math. 24, 51–56, 1968.
- [8] K. Demirci, S. Yildiz and F. Dirik, Approximation via power series method in twodimensional weighted spaces, Bull. Malays. Math. Sci. Soc. 43 (6), 3871–3883, 2020.
- [9] L. Fejér, Untersuchungen über Fouriersche Reihen, Math. Annalen 58, 51–69, 1904.
- [10] T. Y. Gokcer and O. Duman, Regular summability methods in the approximation by max-min operators, Fuzzy Sets and Systems, 426, 106–120, 2022.
- [11] G. H. Hardy, Divergent Series, Oxford, at the Clarendon Press, 1949.
- [12] L. V. Kantorovich, Sur certains développements suivant les polynômes de la forme de S. Bernstein, I. C. R. Acad. Sci. URSS , 563–568, 1930.
- [13] T. A. Keagy and W. F. Ford, Acceleration by subsequence transformations, Pacific J. Math. 132 (2), 357–362, 1988.
- [14] G. G. Lorentz, Bernstein Polynomials, Mathematical Expositions, No. 8. University of Toronto Press, Toronto, 1953.
- [15] R. N. Mohapatra, Quantitative results on almost convergence of a sequence of positive linear operators, J. Approx. Theory 20 (3), 239–250, 1977.
- [16] G. M. Phillips, Bernstein polynomials based on the q-integers, Ann. Numer. Math. 4 (1-4), 511–518, 1997.
- [17] I. Sakaoglu and C. Orhan, Strong summation process in Lp spaces, Nonlinear Anal. 86, 89–94, 2013.
- [18] D. A. Smith and W. F. Ford, Acceleration of linear and logarithmic convergence, SIAM J. Numer. Anal. 16 (2), 223–240, 1979.
- [19] E. Tas and T. Yurdakadim, Approximation by positive linear operators in modular spaces by power series method, Positivity, 21 (4), 1293–1306, 2017.
- [20] M. Unver and C. Orhan, Statistical convergence with respect to power series methods and applications to approximation theory, Numer. Funct. Anal. Optim. 40 (5), 535– 547, 2019.
- [21] E. Voronovskaja, Détermination de la forme asymptotique d’approximation des fonctions par les polynômes de M. Bernstein, Dokl. Akad. Nauk SSSR, 79–85, 1932.
- [22] J. Wimp, Sequence Transformations and Their Applications, Math. Sci. Eng. Vol. 154, Academic Press, New York-London, 1981.
There are 22 citations in total.
Details
| Primary Language | English |
|---|---|
| Subjects | Approximation Theory and Asymptotic Methods |
| Journal Section | Mathematics |
| Authors | |
| Early Pub Date | January 27, 2025 |
| Publication Date | June 24, 2025 |
| Submission Date | May 20, 2024 |
| Acceptance Date | September 22, 2024 |
| Published in Issue | Year 2025 |