Approximation with an arbitrary order by generalized Kantorovich-type and Durrmeyer-type operators on [0;+\infinit)

Sorin Trifa

Abstract


Given an arbitrary sequence ¸n > 0, n 2 N, with the prop-
erty that limn!1 ¸n = 0 as fast we want, in this note we introduce
modi¯ed/generalized Sz¶asz-Kantorovich, Baskakov- Kantorovich, Sz¶asz-
Durrmeyer-Stancu and Baskakov-Sz¶asz-Durrmeyer-Stancu operators in
such a way that on each compact subinterval in [0;+1) the order of
uniform approximation is !1(f;
p
¸n). These modi¯ed operators uni-
formly approximate a Lipschitz 1 function, on each compact subinterval
of [0;1) with the arbitrary good order of approximation
p
¸n. The re-
sults obtained are of a de¯nitive character (that is are the best possible)
and also have a strong unifying character, in the sense that for vari-
ous choices of the nodes ¸n, one can recapture previous approximation
results obtained for these operators by other authors.


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References


Agratini, O., Approximation by Linear Operators (Romanian), University

Press, "Babe»s-Bolyai" University, Cluj-Napoca, 2000.

Baskakov, V. A., An example of a sequence of linear positive operators in the space of continuous functions (Russian), Dokl. Akad. Nauk SSSR, 113(1957), 249-251.

Boehme, T. K., Bruckner, A. M., Functions with convexx means, Paci¯c J.

Math., 14(1964), 1137-1149.

Dieudonn¶e, J., ¶ El¶ements dAnalyse ; 1. Fondements de l'Analyse Moderne, Gauthiers Villars, Paris, 1968.

Djebali, S., Uniform continuity and growth of real continuous functions, Int. J. Math. Education in Science and Technology, 32(2001), No. 5, 677-689.

Gal, S. G., Approximation with an arbitrary order by generalized Sz¶asz-

Mirakjan operators, Stud. Univ. Babes-Bolyai, ser. Math., 59(1)(2014), 77-81.

Gal S. G., Gupta, V., Approximation by complex Sz¶asz-Durrmeyer operators in compact disks, Acta Math. Scientia, 34B(4)(2014), 1157-1165.

Gal, S. G., Gupta, V., Approximation by complex Sz¶asz-Mirakjan-Stancu-Durrmeyer operators in compact disks under exponential growth, Filomat, 29(5)(2015), 1127-1136.

Gal, S. G., Opri»s, D. B., Approximation with an arbitrary order by modified Baskakov type operators, Appl. Math. Comp., 265(2015), 329-332.

Gal, S. G., Opri»s, D. B., Approximation of analytic functions with an arbitrary order by generalized Baskakov-Faber operators in compact sets, Complex Anal. Oper. Theory, 10(2016), No. 2, 369-377.

Gupta, V., Complex Baskakov-Sz¶asz operators in compact semi-disks,

Lobachevskii J. Math., 35(2)(2014), 65-73.

Gupta, V., Overconvergence of complex Baskakov-Sz¶asz-Stancu operators,

Mediterr. J. Math., 12(2)(2015), 455-470.

Gupta, V., Overconvergence of complex Baskakov-Sz¶asz-Stancu operators, Mediterr. J. Math., 12(2)(2015), 455-470.

Mazhar, S. M., Totik V., Approximation by modi¯ed Sz¶asz operators, Acta Sci. Math., 49(1985), 257-269.

Shisha, O., Mond, B., The degree of convergence of linear positive operators, Proc. Nat. Acad. Sci. U.S.A., 60(1968), 1196-1200.

Totik, V., Approximation by Sz¶asz-Mirakjan-Kantorovich operators in Lp (p > 1) (in Russian), Analysis Math., 9(2)(1983), 147-167.

Walczak, Z., On approximation by modified Sz¶asz-Mirakjan operators, Glasnik Mat., 37(2)(2002), 303-319




DOI: http://dx.doi.org/10.24193/subbmath.2017.4.07

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