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Abstract
According to the Weierstrass Approximation Theorem, any continuous function on the closed and bounded interval can be approximated by polynomials. A constructive proof of this theorem uses the so-called Bernstein polynomials. For the approximation of integrable functions, we may consider Kantorovich operators as certain modifications for Bernstein polynomials. In this paper, we investigate the behaviour of Kantorovich operators in Lebesgue spaces. We first give an alternative proof of the uniform boundedness of Kantorovich operators in Lebesgue spaces by using the Riesz-Thorin Interpolation Theorem. In addition, we examine the convergence of Kantorovich operators in the space of essentially bounded functions. We also give an example related to the rate of convergence of Kantorovich operators in a subspace of Lebesgue spaces.
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References
- Arsana, M. P., Gunadi, R., Hakim, D. I., and Sawano, Y., ”On the convergence of Kantorovich operators in Morrey Spaces”, Research Square (2022).
- Bernstein, S., ”D´emonstration du th´eor`eme de Weierstrass fond´ee sur le calcul des probabilit´es”, Comm. Soc. Math. Kharkov 13 (1912), 1-2.
- Burenkov, V., Ghorbanalizadeh, A., and Sawano, Y., ”Uniform boundedness of Kantorovich operators in Morrey spaces”, Positivity 22(4) (2018), 1097-1107.
- Kantorovich, L. V., ”Sur certains d´eveloppements suivant les polynˆomes de la forme de S. Bernstein I, II”, C. R. Acad. Sci. USSR (1930), 563-568 and 595-600.
- Lorentz, G. G., Bernstein Polynomials (2nd ed.), AMS Chelsea Publishing, 1986.
- Maier, V., ”Lp-approximation by Kantoroviˇc operators”, Analysis Mathematica 4(4) (1978), 289-295.
- Zeren, Y., Ismailov, M. and Karacam, C., ”The analogs of the Korovkin theorems in Banach function spaces”, Positivity 26(28) (2018)
References
Arsana, M. P., Gunadi, R., Hakim, D. I., and Sawano, Y., ”On the convergence of Kantorovich operators in Morrey Spaces”, Research Square (2022).
Bernstein, S., ”D´emonstration du th´eor`eme de Weierstrass fond´ee sur le calcul des probabilit´es”, Comm. Soc. Math. Kharkov 13 (1912), 1-2.
Burenkov, V., Ghorbanalizadeh, A., and Sawano, Y., ”Uniform boundedness of Kantorovich operators in Morrey spaces”, Positivity 22(4) (2018), 1097-1107.
Kantorovich, L. V., ”Sur certains d´eveloppements suivant les polynˆomes de la forme de S. Bernstein I, II”, C. R. Acad. Sci. USSR (1930), 563-568 and 595-600.
Lorentz, G. G., Bernstein Polynomials (2nd ed.), AMS Chelsea Publishing, 1986.
Maier, V., ”Lp-approximation by Kantoroviˇc operators”, Analysis Mathematica 4(4) (1978), 289-295.
Zeren, Y., Ismailov, M. and Karacam, C., ”The analogs of the Korovkin theorems in Banach function spaces”, Positivity 26(28) (2018)