Abstract
The paper aims to study a generalization of Szász-Mirakyan-type operators such that their construction depends on a function ρ by using two sequences of functions. To show how the function ρ play a crucial role in the design of the operator, we reconstruct the mentioned operators which preserve exactly two test functions from the set \(\left \{ 1,\rho ,\rho ^{2}\right \}\). We show that these operators provide weighted uniform approximation over unbounded interval. We establish the degree of approximation in terms of a weighted moduli of smoothness associated with the function ρ. Also a Voronovskaya type result is presented. Finally some graphical examples of the mentioned operators are given. Our results show that mentioned operators are sensitive or flexible to point of wive of the rate of convergence to f, depending on our selection of ρ.
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References
Acar, T., Aral, A., Rasa, I.: Modified bernstein-durrmeyer operators. Gen. Math. 22(1), 27–41 (2014)
Altomare, F., Campiti, M.: Korovkin-type approximation theory and its applications, vol. 17. Walter de Gruyter (1994)
Aral, A., Inoan, D., Raṡa, I.: On the generalized Szász–Mirakyan operators. RM 65(3-4), 441–452 (2014)
Braica, P., Piscoran, L., Indrea, A.: Graphical lectures of some king type operators. Acta Univ. Apulensis 34, 163–171 (2013)
Braica, P., Pop, O., Indrea, A.: About a king type operator. Appl. Math. Sci. 6, 145–148 (2012)
Cárdenas-Morales, D., Garrancho, P., Raṡa, I.: Bernstein-type operators which preserve polynomials. Comput. Math. Appl. 62(1), 158–163 (2011)
Duman, O., Özarslan, M.: Szász–Mirakjan type operators providing a better error estimation. Appl. Math. Lett. 20(12), 1184–1188 (2007)
Gadjiev, A.: A problem on the convergence of a sequence of positive linear operators on unbounded sets and theorems that are analogous to PP korovkin’s theorem. Dokl. Akad. Nauk SSSR 218, 1001–1004 (1974)
Holhos, A.: Quantitative estimates for positive linear operators in weighted spaces. General Math. 16(4), 99–110 (2008)
I., B.P.: Operatori de tip king şi aplicatii. Ph.D. School of BaiaMare, Ph.D. thesis (2011)
King, J.: Positive linear operators which preserve x 2. Acta Math. Hungar. 99 (3), 203–208 (2003)
Pop, O., Barbosu, D., Braica, P.: Bernstein-type operators which preserve exactly two test functions. Stud. Sci. Math. Hung. 50(4), 393–405 (2013)
Szasz, O.: Generalization of S. Bernstein’s polynomials to the infinite interval. J. Res. Nat. Bur. Standards 45, 239–245 (1950)
Ulusoy, G., Acar, T.: q-Voronovskaya type theorems for q-Baskakov operators. Math. Methods Appl. Sci., n/a–n/a (2015). doi:10.1002/mma.3784
Weierstrass, K.: Über die analytische darstellbarkeit sogenannter willkürlicher functionen einer reellen veränderlichen. Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin 2, 633–639 (1885)
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Aral, A., Ulusoy, G. & Deni̇z, E. A new construction of Szász-Mirakyan operators. Numer Algor 77, 313–326 (2018). https://doi.org/10.1007/s11075-017-0317-x
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DOI: https://doi.org/10.1007/s11075-017-0317-x