Abstract
We prove Korovkin type results for linear positive operators on \(C\left[ a,b \right] \) with respect to the uniform convergence at a point and some scale functions. Various concrete examples are given.
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References
Altomare, F.: Korovkin-type theorems and approximation by positive linear operators. Surv. Approx. Theory 5, 92–164 (2010)
Altomare, F., Campiti, M.: Korovkin-Type Approximation Theory and Its Applications. de Gruyter Studies in Mathematics, vol. 17. Walter de Gruyter & Co., Berlin (1994)
Bauer, H.: Approximation and abstract boundaries. Am. Math. Mon. 85, 632–647 (1978)
Chittenden, E.W.: Relatively uniform convergence of sequences of functions. Trans. Am. Math. Soc. 15, 197–201 (1914)
Chittenden, E.W.: On the limit functions of sequences of continuous functions converging relatively uniformly. Trans. Am. Math. Soc. 20, 179–184 (1919)
Demirci, K., Boccuto, A., Yıldız, S., Dirik, F.: Relative uniform convergence of a sequence of functions at a point and Korovkin-type approximation theorems. Positivity 24(1), 1–11 (2020)
DeVore, R.A., Lorentz, G.G.: Constructive Approximation. Springer, Berlin (1993)
Klippert, J., Williams, G.: Uniform convergence of a sequence of functions at a point. Int. J. Math. Educ. Sci. Technol. 33, 51–58 (2002)
Korovkin, P.P.: On convergence of linear positive operators in the space of continuous functions. Doklady Akademii Nauk 90, 961–964 (1953) (Russian)
Korovkin, P.P.: Linear Operators and Approximation Theory. Hindustan Publ. Corp, New Delhi (1960)
Lomeli, H.E., Garcia, C.L.: Variations on a theorem of Korovkin. Am. Math. Mon. 113, 744–750 (2006)
Lorentz, G.G.: Inequalities and saturation classes of Bernstein polynomials. In: On Approximation Theory. Proceedings of the Conference at Oberwolfach 1963, pp. 200–207. Birkhaüser Verlag, Basel (1964)
Lorentz, G.G.: Approximation of Functions. Athena Series. Selected Topics in Mathematics, Holt, Rinehart and Winston, New York (1966)
Moore, E.H.: An Introduction to a Form of General Analysis. The New Haven Mathematical Colloquium, Yale University Press, New Haven (1910)
Niculescu, C.P.: An overview of absolute continuity and its applications. In: Inequalities and Applications. International Series of Numerical Mathematics 157, pp. 201–214. Birkhäuser, Basel (2009)
Popa, D.: A multivariate convergence theorem for positive sequences of functions. Arch. Math. 111(1), 61–69 (2018)
Popa, D.: Approximation of the continuous functions on \(l_{p}\) spaces with \(p\) an even natural number. Positivity 24(4), 1135–1149 (2020)
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We would like to thank the reviewer of our paper for carefully reading the manuscript and for such constructive comments, remarks and suggestions which helped improving the first version of the paper.
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Popa, D. Korovkin type results for the uniform convergence at a point. Positivity 25, 1631–1649 (2021). https://doi.org/10.1007/s11117-021-00835-4
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DOI: https://doi.org/10.1007/s11117-021-00835-4
Keywords
- Korovkin theorem
- Uniform convergence at a point
- Asymptotic approximations
- Bernstein–Voronovskaja type approximation