Preserving properties of some Szasz-Mirakyan type operators
DOI:
https://doi.org/10.33993/jnaat531-1408Keywords:
Szasz-Mirakyan type operators, positive linear operator, shape preserving propertiesAbstract
For a family of Szasz-Mirakyan type operators we prove that they preserve convex-type functions and that a monotonicity property verified by Cheney and Sharma in the case Szasz-Mirakyan operators holds for the variation study here. We also verify that several modulus of continuity are preserved.
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U. Abel, M. Ivan and X. M. Zeng, Asymptotic expansion for Szasz-Mirakyan operators, AIP Conference Proceedings, 936 (2007), 79-782. DOI: https://doi.org/10.1063/1.2790269
https://doi.org/0.1063/1.2790269
T. Acar, A. Aral and I. Rasa, The new forms of Voronovskaya’s theorem in
weighted spaces, Positivity, 20 (1) (2015), 25-40.
https://doi.org/10.1007/s11117-015-0338-4 DOI: https://doi.org/10.1007/s11117-015-0338-4
J. A. Adell and A. Lekuona, Best constants in preservation of global smoothness for Szasz-Mirakyan operators, J. Math. Anal. Appl. 338 (2008), 753-757.
https://doi.org/10.1016/j.jmaa.2007.05.064 DOI: https://doi.org/10.1016/j.jmaa.2007.05.064
N. T. Amanov, On the uniform weighted approximation by Szasz–Mirakjan operators, Analysis Mathematica, 18 (1992), 167-184. https://doi.org/10.1007/BF01911084 DOI: https://doi.org/10.1007/BF01911084
M. Becker, Global approximation theorems for Szasz-Mirakjan and Baskakov operators in polynomial weight spaces, Indiana Univ. Math. J., 27 (1)(1978), 127-142.
J. Bustamante and C. Castaneda Roldan, Direct and inverse results in Holder norms, J. Approx. Theory, 138 (2006) 112-123.
https://doi.org/10.1016/j.jat.2005.10.004 DOI: https://doi.org/10.1016/j.jat.2005.10.004
J. Bustamante, J. M. Quesada and L. Morales de la Cruz, Direct estimate for positive linear operators in polynomial weighted spaces, J. Appr. Theory, 162 (2010),
-1508. url https://doi.org/10.1016/j.jat.2010.04.001 DOI: https://doi.org/10.1016/j.jat.2010.04.001
J. Bustamante, A. Carrillo-Zentella and J. M. Quesada, Direct and strong converse theorems for a general sequence of positive linear operators, Acta Math. Hungar. 136 (1-2) (2012), 90-106.
https://doi.org/10.1007/s10474-012-0196-5 DOI: https://doi.org/10.1007/s10474-012-0196-5
E. Cheney and A. Sharma, Bernstein power series, Canad. J. Math., 16 (1964), 241-252. DOI: https://doi.org/10.4153/CJM-1964-023-1
M. Chu, On the Szasz operators Voronoskaja type theorem, J. Anhui Normal Univ. (Nat. Sci.), 18 (1) (1995), 20-23.
R. A. DeVore and G. G. Lorentz, Constructive Approximation, Springer-Verlag, Berlin Heidelberg New York, (1993). DOI: https://doi.org/10.1007/978-3-662-02888-9
H. Dong and Q. Qi, Shape preserving properties of parametric Szasz type operators on unbounded intervals, Symmetry, 15 (2023), 1755.
https://doi.org/10.3390/sym15091755 DOI: https://doi.org/10.3390/sym15091755
T. Hermann, On the Szasz-Mirakian operators, Acta Math. Acad, Sci. Hungar, 32 (1-2), (1978), 163-173.
https://doi.org/10.1007/BF01902211 DOI: https://doi.org/10.1007/BF01902211
I. Horova, Linear positive operators of convex functions, Mathematica, 10 (33) (1968), 275-283.
X.Q. Hou and Y.C. Xue, On the property of some linear positive operators preserving the class Λω (A), J. Ningxia Univ. (Nat. Sci. Ed.) 16 (1995), 11-16 (in Chinese)
N. ̇Ispir, On Modified Baskakov operators on weighted spaces, Turk. J. Math, 25 (2001), 355-365.
G. Jiang, On the inverse theorem for Szasz-Mirakjan operators, Liupanshui Normal Univ., 4 (1993), 8-10 (in Chinese).
M. K. Khan, B. Della Vecchia, and A. Fassih, On the monotonicity of positive linear operators, J. Approx. Theory 92 (1998), 22-37. DOI: https://doi.org/10.1006/jath.1996.3113
W. Kratz and U. Stadtm ̈uller, On the uniform modulus of continuity of certain discrete approximation operators, J. Approx. Theory, 54 (1988), 326-337.
https://doi.org/10.1016/0021-9045(88)90009-3 DOI: https://doi.org/10.1016/0021-9045(88)90009-3
C. Li and Y. Zhao, Weighted approximation with Szasz-Mirakjan operators, Acta Scie. Natur. Univ. Pekinensis 37 (1) (2001), 6–11 (in Chinese).
L. Liu, Y. Xue and W. Sun, Strong converse inequalities for Szasz-Mirakjian operators with weights, Chi. Quart. Math., 23 (3) (2008), 384-389.
L. Liu, G. Yang, and S. Guo, Strong converse inequality for Szasz operators, J. Math. Research Expo., 28 (1) (2008), 147–155.
A.J. Lopez-Moreno, Weighted simultaneous approximation with Baskakov type operators, Acta Math. Hungar., 104 (1–2) (2004), 143-151.
https://doi.org/10.1023/B:AMHU.0000034368.81211.23 DOI: https://doi.org/10.1023/B:AMHU.0000034368.81211.23
A. Lupas, Some properties of the linear positive operators (I), Mathematica Cluj, 9 (1967), 77-83.
D. Miclaus and O. T. Pop, The generalization of certain results for Szasz-Mirakjan-Schurer operators, Creat. Math. Inform. 21 (1) (2012), 79-85. DOI: https://doi.org/10.37193/CMI.2012.01.10
F. Schurer, Linear positive operators in approximation theory, Math. Inst. Techn., Univ. Delft Report, 1962
F. Schurer, On Linear positive operators in approximation theory, Delft University of Technology, Delft, 1965
P. C. Sikkema, On some linear positive operators, Indag. Math. 32 (1970), 327-337. DOI: https://doi.org/10.1016/S1385-7258(70)80037-3
P. C. Sikkema, Uber die Schurerschen linearen positiven Operatoren I, Indag. Math. 78 (3) (1975), 230-242. DOI: https://doi.org/10.1016/1385-7258(75)90037-2
O. Szasz, Generalization of S. Bernstein’s polynomials to the infinite interval, J. Res. Nat. Bur. Standards, 45 (1950), 239-245.
https://doi.org/10.6028/JRES.045.024 DOI: https://doi.org/10.6028/jres.045.024
B. D. Vechia, On the preservation of Lipschitz constants for some linear operators, Bollettino Un. Mat. Ita., (16) (1) (1989), 125-136.
X. Wang, On the proof of a theorem for Szasz-Mirakjan operators, J. Hangzhou Univ. (Nat. Sci.), 19 (2) (1992), 139-143 (in Chinese).
X. Wang, X. Li, W. Wang and S. Ding, Equivalent description on derivatives of Szasz-Mirakjan operators, J. South. West Univ. (Nat. Sci.), 8 (2011), 115-118 (in Chinese).
L. Xie, On direct theorems for Szasz-Mirakian operators, J. Lishui Teach. Colle., 17 (2) (1995), 1-2 (in Chinese).
L. Xie, Inverse theorems for Szasz-Mirakjian operators, J. Lishui Teach. Colle., 22 (2) (2000) 1-3 (in Chinese).
Y. Xin, C. Li and Q. Gao, Approximation qualities for the iterated Boolean sums of Szasz operators, J. Hebei Normal Univ., 32 (6) (2008), 713-717 (in Chinese).
Z. Zhao, On the property of Szasz-Mirakyan polynomials preserving the monotonicity, J. Gansu Edu. Coll., 15 (3) (2001), 1-3 (in Chinese).
L. Zhen, The shape preserving property and approximation order of Szasz-Mirakjan operators, J. Hunan Univ. Tech., 25 (2) (2011), 5-9 (in Chinese).
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