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On asymptotic formulae via summability. (English) Zbl 1227.40006
The authors provide some Korovkin-type result on asymptotic formulae for sequences of linear operators which are A-summable.
MSC:
40J05Summability in abstract structures
41A36Approximation by positive operators
References:
[1]Aguilera, F.; Cárdenas-Morales, D.; Garrancho, P.; Hernández, J. M.: Quantitative results in conservative approximation via summability, Automat. comput. Appl. math. 17, No. 2, 201-208 (2008)
[2]Altomare, F.; Campiti, M.: Korovkin-type approximation theory and its applications, (1994)
[3]Bell, H. T.: Order summability and almost convergence, Proc. am. Math. soc. 38, 548-552 (1973) · Zbl 0259.40003 · doi:10.2307/2038948
[4]Cárdenas-Morales, D.; Garrancho, P.; Muñoz-Delgado, F. J.: A result of asymptotic formulae for linear k-convex operators, Int. J. Differ. equ. Appl. 2, No. 3, 335-347 (2001) · Zbl 1040.41015
[5]Cárdenas-Morales, D.; Muñoz-Delgado, F. J.: A Korovkin-type result in ck, an application to the mn operators, Approx. theory appl. (N. S.) 17, No. 3, 1-13 (2001) · Zbl 0995.41011 · doi:10.1023/A:1015505023069
[6]Devore, R. A.: The approximation of continuous functions by positive linear operators, lectures notes in mathematics 293, (1972) · Zbl 0276.41011
[7]Duman, O.; Orhan, C.: Rates of A-statistical convergence of positive linear operators, Appl. math. Lett. 18, 1339-1344 (2005) · Zbl 1085.41012 · doi:10.1016/j.aml.2005.02.029
[8]Eisenberg, S.; Wood, B.: Approximation of analytic functions by Bernstein-type operators, J. approx. Theory 6, 242-248 (1972) · Zbl 0242.30039 · doi:10.1016/0021-9045(72)90055-X
[9]Fast, H.: Sur la convergence statistique, Colloq. math. 2, 241-244 (1951) · Zbl 0044.33605
[10]King, J. P.: The lototsky transform and Bernstein polynomials, Can. J. Math. 18, 89-91 (1966) · Zbl 0134.05101 · doi:10.4153/CJM-1966-011-1
[11]King, J. P.; Swetits, J. J.: Positive linear operators and summability, Aust. J. Math. 11, 281-291 (1970) · Zbl 0199.45101 · doi:10.1017/S1446788700006650
[12]Korovkin, P. P.: Linear operators and approximation theory, (1960)
[13]Lorentz, G. G.: A contribution to the theory of divergent sequences, Acta. math. 80, 167-190 (1948) · Zbl 0031.29501 · doi:10.1007/BF02393648
[14]Mohapatra, R. N.: Quantitative results on almost convergence of a sequence of positive linear operators, J. approx. Theory 20, 239-250 (1977) · Zbl 0351.41010 · doi:10.1016/0021-9045(77)90058-2
[15]Muñoz-Delgado, F. J.; Cárdenas-Morales, D.: Almost convexity and quantitative Korovkin type results, Appl. math. Lett. 11, No. 4, 105-108 (1998) · Zbl 0942.41013 · doi:10.1016/S0893-9659(98)00065-2
[16]Swetits, J. J.: On summability and positive linear operators, J. approx. Theory 25, 186-188 (1979) · Zbl 0422.41019 · doi:10.1016/0021-9045(79)90008-X