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On uniform approximation by some classical Bernstein-type operators. (English) Zbl 1048.41013
Summary: We investigate the functions for which certain classical families of operators of probabilistic type over noncompact intervals provide uniform approximation on the whole interval. The discussed examples include the Szász operators, the Szász-Durrmeyer operators, the gamma operators, the Baskakov operators, and the Meyer-König and Zeller operators. We show that some results of Totik remain valid for unbounded functions, at the same time that we give simple rates of convergence in terms of the usual modulus of continuity. We also show by a counterexample that the result for Meyer-König and Zeller operators does not extend to Cheney and Sharma operators.

41A35Approximation by operators (in particular, by integral operators)
Full Text: DOI
[1] Adell, J. A.; De La Cal, J.: Bernstein--Durrmeyer operators. Comput. math. Appl. 30, 1-14 (1995) · Zbl 0839.41018
[2] Adell, J. A.; De La Cal, J.: Bernstein-type operators diminish the ${\phi}$-variation. Constr. approx. 12, 489-507 (1996) · Zbl 0873.41006
[3] Adell, J. A.; De La Cal, J.; Pérez-Palomares, A.: On the cheney and Sharma operator. J. math. Anal. appl. 200, 663-679 (1996) · Zbl 0857.41020
[4] Cheney, E. W.; Sharma, A.: Bernstein power series. Canad. J. Math. 16, 241-252 (1964) · Zbl 0128.29001
[5] Lupas, A.; Müller, M.: Approximationseigenschaften der gammaoperatoren. Math. Z. 98, 208-226 (1967) · Zbl 0171.02301
[6] Mazhar, S. M.; Totik, V.: Approximation by modified szász operators. Acta sci. Math. 49, 257-269 (1985) · Zbl 0611.41013
[7] Totik, V.: Uniform approximation by szász--mirakjan type operators. Acta math. Hungar. 41, 291-307 (1983) · Zbl 0513.41013
[8] Totik, V.: Uniform approximation by baskakov and Meyer--könig and zeller operators. Period. math. Hungar. 14, 209-228 (1983) · Zbl 0497.41015
[9] Totik, V.: Uniform approximation by positive operators on infinite intervals. Anal. math. 10, 163-182 (1984) · Zbl 0579.41015