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Statistical approximation properties of high order operators constructed with the Chan-Chyan-Srivastava polynomials. (English) Zbl 1236.41018

The authors introduce a more general positive linear operator than that studied in [E. Erkus, O. Duman and H. M. Srivastava, Appl. Math. Comput. 182, No. 1, 213–222 (2006; Zbl 1103.41024)], by involving the derivatives of the function to be approximated over an appropriate space. Korovkin-type approximation result for the foregoing more general positive linear operator has been studied in the sense of \(A-\)statistical convergence see H. Fast, [Colloq. Math. 2, 241–244 (1951; Zbl 0044.33605)], which is more general than the usual Cauchy convergence.

MSC:

41A25 Rate of convergence, degree of approximation
41A35 Approximation by operators (in particular, by integral operators)
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[1] Altın, A.; Aktaş, R.; Erkuş-Duman, E., On a multivariable extension for the extended Jacobi polynomials, J. Math. Anal. Appl., 353, 121-133 (2009) · Zbl 1172.33002
[2] Anastassiou, G. A.; Duman, O., A Baskakov type generalization of statistical Korovkin theory, J. Math. Anal. Appl., 340, 476-486 (2008) · Zbl 1133.41004
[3] Boos, J., Classical and Modern Methods in Summability (2000), Oxford University Press: Oxford University Press UK · Zbl 0954.40001
[4] Chan, W.-C. C.; Chyan, C.-J.; Srivastava, H. M., The Lagrange polynomials in several variables, Integral Transform. Spec. Funct., 12, 139-148 (2001) · Zbl 1057.33003
[5] Chen, K.-Y.; Liu, S.-J.; Srivastava, H. M., Some new results for the Lagrange polynomials in several variables, ANZIAM J., 49, 243-258 (2007) · Zbl 1148.33002
[6] Doğru, O.; Duman, O.; Orhan, C., Statistical approximation by generalized Meyer-König and Zeller type operators, Stud. Sci. Math. Hungar., 40, 359-371 (2003) · Zbl 1065.41040
[7] Duman, O., Higher order generalization of positive linear operators defined by a class of Borel measures, Turkish J. Math., 31, 333-339 (2007) · Zbl 1131.41306
[8] Duman, O., A-Statistical convergence of sequences of convolution operators, Taiwanese J. Math., 12, 523-536 (2008) · Zbl 1348.41019
[9] Duman, O.; Erkuş, E.; Gupta, V., Statistical rates on the multivariate approximation theory, Math. Comput. Model., 44, 763-770 (2006) · Zbl 1132.41330
[10] Duman, O.; Khan, M. K.; Orhan, C., \(A\)-Statistical convergence of approximating operators, Math. Inequal. Appl., 6, 689-699 (2003) · Zbl 1086.41008
[11] Duman, O.; Özarslan, M. A.; Aktuğlu, H., Better error estimation for Szász-Mirakjan-Beta operators, J. Comput. Anal. Appl., 10, 53-59 (2008) · Zbl 1134.41010
[12] Erdélyi, A.; Magnus, W.; Oberhettinger, F.; Tricomi, F. G., Higher Transcendental Functions, vol. 3 (1955), McGraw-Hill Book Company: McGraw-Hill Book Company New York, Toronto and London · Zbl 0064.06302
[13] Erkuş, E.; Duman, O.; Srivastava, H. M., Statistical approximation of certain positive linear operators constructed by means of the Chan-Chyan-Srivastava polynomials, Appl. Math. Comput., 182, 213-222 (2006) · Zbl 1103.41024
[14] Erkuş-Duman, E.; Duman, O., Integral type generalizations of operators obtained from certain multivariate polynomials, Calcolo, 45, 53-67 (2008) · Zbl 1142.41005
[15] Fast, H., Sur la convergence statistique (in French), Colloq. Math., 2, 241-244 (1951) · Zbl 0044.33605
[16] Freedman, A. R.; Sember, J. J., Densities and summability, Pac. J. Math., 95, 293-305 (1981) · Zbl 0504.40002
[17] Fridy, J. A., On statistical convergence, Analysis, 5, 301-313 (1985) · Zbl 0588.40001
[18] Gadjiev, A. D.; Orhan, C., Some approximation theorems via statistical convergence, Rocky Mountain J. Math., 32, 129-138 (2002) · Zbl 1039.41018
[19] Karakuş, S.; Demirci, K.; Duman, O., Equi-statistical convergence of positive linear operators, J. Math. Anal. Appl., 339, 1065-1072 (2008) · Zbl 1131.41008
[20] Kirov, G. H.; Popova, L., A generalization of the linear positive operators, Math. Balkanica, 7, 149-162 (1993) · Zbl 0833.41016
[21] Liu, S.-J.; Chyan, C.-J.; Lu, H.-C.; Srivastava, H. M., Multiple integral representations for some families of hypergeometric and other polynomials, Math. Comput. Model. (2011), in press, DOI: 10.1016/j.mcm.2011.04.01
[22] Miller, H. I., A measure theoretical subsequence characterization of statistical convergence, Trans. Am. Math. Soc., 347, 1811-1819 (1995) · Zbl 0830.40002
[23] Özarslan, M. A.; Duman, O., Approximation theorems by Meyer-König and Zeller type operators, Chaos, Solitons Fractals, 41, 451-456 (2009) · Zbl 1198.41014
[24] Özarslan, M. A.; Duman, O.; Doğru, O., Rates of A-statistical convergence of approximating operators, Calcolo, 42, 93-104 (2005) · Zbl 1104.41018
[25] Radu, C., On statistical approximation of a general class of positive linear operators extended in q-calculus, Appl. Math. Comput., 215, 2317-2325 (2009) · Zbl 1179.41025
[26] Srivastava, H. M.; Manocha, H. L., A Treatise on Generating Functions, Halsted Press (Ellis Horwood Limited, Chichester) (1984), John Wiley and Sons: John Wiley and Sons New York, Chichester, Brisbane and Toronto · Zbl 0535.33001
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