Abstract
n the present paper, we propose modification of the exponential type operators, which are connected with \(x^3\). Such operators are connected with exponential functions. We estimate moments and establish some direct results in terms of modulus of continuity.
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References
Aral, A., Otrocol, D., Rasa, I.: On approximation by some Bernstein–Kantorovich exponential–type polynomials. Period Math. Hungar. https://doi.org/10.1007/s10998-019-00284-3
Acar, T., Aral, A., Rasa, I.: Positive linear operators preserving \(\tau \) and \(\tau ^2\). Constr. Math. Anal. 2(3), 98–102 (2019)
Aral, A., Limmam, M.L., Ozsarac, F.: Approximation properties of Sz \(\cdot \)z Mirakyan–Kantorovich type operators. Math. Methods Appl. Sci. 42(16), 5233–5240 (2019)
Boyanov, B.D., Veselinov, V.M.: A note on the approximation of functions in an infinite interval by linear positive operators. Bull. Math. Soc. Sci. Math. Roum 14(62), 9–13 (1970)
Gadziev, A.D.: Theorems of the type of P. P. Korovkin’s theorems. Mat. Zametki 20(5), 781–786 (1976)
Gupta, V.: Approximation with certain exponential operators, Revista de la Real Academia de Ciencias Exactas. Físicas y Naturales. Serie A. Matemáticas 114(2), 51 (2020)
Gupta, V., Agarwal, R.P.: Convergence Estimates in Approximation Theory. Springer, Cham (2014)
Gupta, V., Tachev, G.: Approximation with Positive Linear Operators and Linear Combinations, Series: Developments in Mathematics, vol. 50. Springer, London (2017)
Ismail, M., May, C.P.: On a family of approximation operators. J. Math. Anal. Appl. 63, 446–462 (1978)
King, J.P.: Positive linear operators which preserve \(x^2,\). Acta. Math. Hungar. 99(3), 203–208 (2003)
Ozsarac, F., Acar, T.: Reconstruction of Baskakov operators preserving some exponential functions. Math. Methods Appl. Sci. 42, 5124–5132 (2019)
Pǎltǎnea, R.: A note on Bernstein–Kantorovich operators. Bull. Univ. Transil. Brasov Ser. III 6(55(2)), 27–32 (2013)
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Gupta, V., Aral, A. & Muraru, CV. Modification of Exponential Type Operators Preserving Exponential Functions Connected with \(x^3\). Mediterr. J. Math. 18, 222 (2021). https://doi.org/10.1007/s00009-021-01851-0
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DOI: https://doi.org/10.1007/s00009-021-01851-0